| L(s) = 1 | − 3·4-s + 7·9-s − 2·11-s + 6·16-s − 6·19-s + 4·29-s − 12·31-s − 21·36-s − 42·41-s + 6·44-s + 10·49-s − 16·59-s − 16·61-s − 10·64-s − 16·71-s + 18·76-s − 44·79-s + 15·81-s + 6·89-s − 14·99-s − 32·101-s − 8·109-s − 12·116-s − 17·121-s + 36·124-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | − 3/2·4-s + 7/3·9-s − 0.603·11-s + 3/2·16-s − 1.37·19-s + 0.742·29-s − 2.15·31-s − 7/2·36-s − 6.55·41-s + 0.904·44-s + 10/7·49-s − 2.08·59-s − 2.04·61-s − 5/4·64-s − 1.89·71-s + 2.06·76-s − 4.95·79-s + 5/3·81-s + 0.635·89-s − 1.40·99-s − 3.18·101-s − 0.766·109-s − 1.11·116-s − 1.54·121-s + 3.23·124-s + 0.0887·127-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 5^{12} \cdot 37^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 5^{12} \cdot 37^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8875501007\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8875501007\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 + T^{2} )^{3} \) |
| 5 | \( 1 \) |
| 37 | \( ( 1 + T^{2} )^{3} \) |
| good | 3 | \( 1 - 7 T^{2} + 34 T^{4} - 107 T^{6} + 34 p^{2} T^{8} - 7 p^{4} T^{10} + p^{6} T^{12} \) |
| 7 | \( 1 - 10 T^{2} + 25 p T^{4} - 1000 T^{6} + 25 p^{3} T^{8} - 10 p^{4} T^{10} + p^{6} T^{12} \) |
| 11 | \( ( 1 + T + 10 T^{2} - 3 T^{3} + 10 p T^{4} + p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 13 | \( 1 - 22 T^{2} + 35 p T^{4} - 6672 T^{6} + 35 p^{3} T^{8} - 22 p^{4} T^{10} + p^{6} T^{12} \) |
| 17 | \( 1 - 59 T^{2} + 1794 T^{4} - 35879 T^{6} + 1794 p^{2} T^{8} - 59 p^{4} T^{10} + p^{6} T^{12} \) |
| 19 | \( ( 1 + 3 T + 32 T^{2} + 35 T^{3} + 32 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 23 | \( 1 - 74 T^{2} + 2559 T^{4} - 62732 T^{6} + 2559 p^{2} T^{8} - 74 p^{4} T^{10} + p^{6} T^{12} \) |
| 29 | \( ( 1 - 2 T + 35 T^{2} - 76 T^{3} + 35 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 31 | \( ( 1 + 6 T + 75 T^{2} + 318 T^{3} + 75 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 41 | \( ( 1 + 21 T + 254 T^{2} + 1937 T^{3} + 254 p T^{4} + 21 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 43 | \( 1 + 62 T^{2} + 6295 T^{4} + 228116 T^{6} + 6295 p^{2} T^{8} + 62 p^{4} T^{10} + p^{6} T^{12} \) |
| 47 | \( 1 - 170 T^{2} + 14895 T^{4} - 852620 T^{6} + 14895 p^{2} T^{8} - 170 p^{4} T^{10} + p^{6} T^{12} \) |
| 53 | \( 1 - 82 T^{2} + 9687 T^{4} - 464476 T^{6} + 9687 p^{2} T^{8} - 82 p^{4} T^{10} + p^{6} T^{12} \) |
| 59 | \( ( 1 + 8 T + 49 T^{2} - 132 T^{3} + 49 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 61 | \( ( 1 + 8 T + 139 T^{2} + 686 T^{3} + 139 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 67 | \( 1 - 215 T^{2} + 19042 T^{4} - 18185 p T^{6} + 19042 p^{2} T^{8} - 215 p^{4} T^{10} + p^{6} T^{12} \) |
| 71 | \( ( 1 + 8 T + 169 T^{2} + 846 T^{3} + 169 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 73 | \( 1 - 107 T^{2} + 17470 T^{4} - 1087847 T^{6} + 17470 p^{2} T^{8} - 107 p^{4} T^{10} + p^{6} T^{12} \) |
| 79 | \( ( 1 + 22 T + 377 T^{2} + 3708 T^{3} + 377 p T^{4} + 22 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 83 | \( 1 + 5 T^{2} + 13334 T^{4} - 117147 T^{6} + 13334 p^{2} T^{8} + 5 p^{4} T^{10} + p^{6} T^{12} \) |
| 89 | \( ( 1 - 3 T + 240 T^{2} - 507 T^{3} + 240 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 97 | \( 1 - 538 T^{2} + 124559 T^{4} - 15864108 T^{6} + 124559 p^{2} T^{8} - 538 p^{4} T^{10} + p^{6} T^{12} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.72322056012569091584678812158, −4.71000602386067210101167595407, −4.51829684238159784141421063304, −4.42356707841982821144043731036, −4.23885117069323885603736832279, −4.13480622137287651965643410387, −4.02925204425138839554718935885, −3.62624179276937420010403737872, −3.61396714046867981297807296580, −3.61322112380738309964549347529, −3.28104480738997440261681114396, −3.16547391109303305961614879876, −2.87976827713540380128001587643, −2.73196048587982389954096408344, −2.63219155795501243909286484836, −2.53413459075016793030043642596, −1.83537041438676053534944797217, −1.74387961788892962886322274658, −1.63336208805406994174518909620, −1.57197146817583227965798154464, −1.50992630095824796413878015376, −1.42400108304827370403077548257, −0.53370545389602944928403882650, −0.49348959718634535225548284348, −0.19037356928711635920329618237,
0.19037356928711635920329618237, 0.49348959718634535225548284348, 0.53370545389602944928403882650, 1.42400108304827370403077548257, 1.50992630095824796413878015376, 1.57197146817583227965798154464, 1.63336208805406994174518909620, 1.74387961788892962886322274658, 1.83537041438676053534944797217, 2.53413459075016793030043642596, 2.63219155795501243909286484836, 2.73196048587982389954096408344, 2.87976827713540380128001587643, 3.16547391109303305961614879876, 3.28104480738997440261681114396, 3.61322112380738309964549347529, 3.61396714046867981297807296580, 3.62624179276937420010403737872, 4.02925204425138839554718935885, 4.13480622137287651965643410387, 4.23885117069323885603736832279, 4.42356707841982821144043731036, 4.51829684238159784141421063304, 4.71000602386067210101167595407, 4.72322056012569091584678812158
Plot not available for L-functions of degree greater than 10.
