| L(s) = 1 | − 4-s + 4·5-s − 9-s + 16-s + 8·19-s − 4·20-s + 11·25-s + 20·29-s − 2·31-s + 36-s + 4·41-s − 4·45-s + 10·49-s + 12·59-s − 16·61-s − 64-s − 16·71-s − 8·76-s − 24·79-s + 4·80-s + 81-s + 28·89-s + 32·95-s − 11·100-s − 4·109-s − 20·116-s − 22·121-s + ⋯ |
| L(s) = 1 | − 1/2·4-s + 1.78·5-s − 1/3·9-s + 1/4·16-s + 1.83·19-s − 0.894·20-s + 11/5·25-s + 3.71·29-s − 0.359·31-s + 1/6·36-s + 0.624·41-s − 0.596·45-s + 10/7·49-s + 1.56·59-s − 2.04·61-s − 1/8·64-s − 1.89·71-s − 0.917·76-s − 2.70·79-s + 0.447·80-s + 1/9·81-s + 2.96·89-s + 3.28·95-s − 1.09·100-s − 0.383·109-s − 1.85·116-s − 2·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 864900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.055100252\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.055100252\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_2$ | \( 1 + T^{2} \) |
| 3 | $C_2$ | \( 1 + T^{2} \) |
| 5 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| 31 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 42 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 78 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 150 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 178 T^{2} + p^{2} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.19162207819299726554368610715, −10.00047568405200539228697810234, −9.274767585490906418972086826201, −9.237516804256183078672722751031, −8.569801162487930724931030941037, −8.537812777193137333475258810287, −7.60075428088826133685036102694, −7.45917870493747497762657713033, −6.63412707880630250998924878427, −6.47160133621760690562630147376, −5.72811720796610628786290294047, −5.70918964423839515412498593212, −4.97314598266374029261205774366, −4.75980100681370702151201742715, −4.11181043515319211873691267605, −3.21238279759276996273909879409, −2.81595747864006454330835131972, −2.41471440131578098084008800367, −1.36574474267146578414953301696, −0.967578865639452462690145627241,
0.967578865639452462690145627241, 1.36574474267146578414953301696, 2.41471440131578098084008800367, 2.81595747864006454330835131972, 3.21238279759276996273909879409, 4.11181043515319211873691267605, 4.75980100681370702151201742715, 4.97314598266374029261205774366, 5.70918964423839515412498593212, 5.72811720796610628786290294047, 6.47160133621760690562630147376, 6.63412707880630250998924878427, 7.45917870493747497762657713033, 7.60075428088826133685036102694, 8.537812777193137333475258810287, 8.569801162487930724931030941037, 9.237516804256183078672722751031, 9.274767585490906418972086826201, 10.00047568405200539228697810234, 10.19162207819299726554368610715
