| L(s) = 1 | − 9·3-s − 56·7-s + 81·9-s + 156·11-s − 350·13-s − 786·17-s + 740·19-s + 504·21-s − 2.37e3·23-s − 729·27-s + 2.57e3·29-s − 4.57e3·31-s − 1.40e3·33-s + 1.22e4·37-s + 3.15e3·39-s − 1.02e4·41-s + 1.60e4·43-s − 864·47-s − 1.36e4·49-s + 7.07e3·51-s + 1.76e4·53-s − 6.66e3·57-s + 4.86e4·59-s − 3.37e4·61-s − 4.53e3·63-s − 3.52e3·67-s + 2.13e4·69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.431·7-s + 1/3·9-s + 0.388·11-s − 0.574·13-s − 0.659·17-s + 0.470·19-s + 0.249·21-s − 0.936·23-s − 0.192·27-s + 0.568·29-s − 0.855·31-s − 0.224·33-s + 1.46·37-s + 0.331·39-s − 0.950·41-s + 1.32·43-s − 0.0570·47-s − 0.813·49-s + 0.380·51-s + 0.863·53-s − 0.271·57-s + 1.82·59-s − 1.16·61-s − 0.143·63-s − 0.0959·67-s + 0.540·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.288434813\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.288434813\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + p^{2} T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 8 p T + p^{5} T^{2} \) |
| 11 | \( 1 - 156 T + p^{5} T^{2} \) |
| 13 | \( 1 + 350 T + p^{5} T^{2} \) |
| 17 | \( 1 + 786 T + p^{5} T^{2} \) |
| 19 | \( 1 - 740 T + p^{5} T^{2} \) |
| 23 | \( 1 + 2376 T + p^{5} T^{2} \) |
| 29 | \( 1 - 2574 T + p^{5} T^{2} \) |
| 31 | \( 1 + 4576 T + p^{5} T^{2} \) |
| 37 | \( 1 - 12202 T + p^{5} T^{2} \) |
| 41 | \( 1 + 10230 T + p^{5} T^{2} \) |
| 43 | \( 1 - 16084 T + p^{5} T^{2} \) |
| 47 | \( 1 + 864 T + p^{5} T^{2} \) |
| 53 | \( 1 - 17658 T + p^{5} T^{2} \) |
| 59 | \( 1 - 48684 T + p^{5} T^{2} \) |
| 61 | \( 1 + 33778 T + p^{5} T^{2} \) |
| 67 | \( 1 + 3524 T + p^{5} T^{2} \) |
| 71 | \( 1 - 38280 T + p^{5} T^{2} \) |
| 73 | \( 1 - 79702 T + p^{5} T^{2} \) |
| 79 | \( 1 - 99248 T + p^{5} T^{2} \) |
| 83 | \( 1 - 22284 T + p^{5} T^{2} \) |
| 89 | \( 1 - 94650 T + p^{5} T^{2} \) |
| 97 | \( 1 + 9122 T + p^{5} T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.96118098093104326530873007478, −9.952535552291571298191118722072, −9.208698581719635193929907173910, −7.924537525662358293759141079535, −6.87126587329669251530244506652, −6.01626035148281228015073941996, −4.87581228705711385635161358352, −3.73805409138570306006798219859, −2.22817158215596158169425795373, −0.64652254359772108403161579292,
0.64652254359772108403161579292, 2.22817158215596158169425795373, 3.73805409138570306006798219859, 4.87581228705711385635161358352, 6.01626035148281228015073941996, 6.87126587329669251530244506652, 7.924537525662358293759141079535, 9.208698581719635193929907173910, 9.952535552291571298191118722072, 10.96118098093104326530873007478
