| L(s) = 1 | − 4·3-s − 25·5-s − 192·7-s − 227·9-s − 148·11-s − 286·13-s + 100·15-s − 1.67e3·17-s + 1.06e3·19-s + 768·21-s − 2.97e3·23-s + 625·25-s + 1.88e3·27-s + 3.41e3·29-s + 2.44e3·31-s + 592·33-s + 4.80e3·35-s − 182·37-s + 1.14e3·39-s − 9.39e3·41-s − 1.24e3·43-s + 5.67e3·45-s + 1.20e4·47-s + 2.00e4·49-s + 6.71e3·51-s − 2.38e4·53-s + 3.70e3·55-s + ⋯ |
| L(s) = 1 | − 0.256·3-s − 0.447·5-s − 1.48·7-s − 0.934·9-s − 0.368·11-s − 0.469·13-s + 0.114·15-s − 1.40·17-s + 0.673·19-s + 0.380·21-s − 1.17·23-s + 1/5·25-s + 0.496·27-s + 0.752·29-s + 0.457·31-s + 0.0946·33-s + 0.662·35-s − 0.0218·37-s + 0.120·39-s − 0.873·41-s − 0.102·43-s + 0.417·45-s + 0.798·47-s + 1.19·49-s + 0.361·51-s − 1.16·53-s + 0.164·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.4335863759\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4335863759\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + p^{2} T \) |
| good | 3 | \( 1 + 4 T + p^{5} T^{2} \) |
| 7 | \( 1 + 192 T + p^{5} T^{2} \) |
| 11 | \( 1 + 148 T + p^{5} T^{2} \) |
| 13 | \( 1 + 22 p T + p^{5} T^{2} \) |
| 17 | \( 1 + 1678 T + p^{5} T^{2} \) |
| 19 | \( 1 - 1060 T + p^{5} T^{2} \) |
| 23 | \( 1 + 2976 T + p^{5} T^{2} \) |
| 29 | \( 1 - 3410 T + p^{5} T^{2} \) |
| 31 | \( 1 - 2448 T + p^{5} T^{2} \) |
| 37 | \( 1 + 182 T + p^{5} T^{2} \) |
| 41 | \( 1 + 9398 T + p^{5} T^{2} \) |
| 43 | \( 1 + 1244 T + p^{5} T^{2} \) |
| 47 | \( 1 - 12088 T + p^{5} T^{2} \) |
| 53 | \( 1 + 23846 T + p^{5} T^{2} \) |
| 59 | \( 1 + 20020 T + p^{5} T^{2} \) |
| 61 | \( 1 + 32302 T + p^{5} T^{2} \) |
| 67 | \( 1 - 60972 T + p^{5} T^{2} \) |
| 71 | \( 1 - 32648 T + p^{5} T^{2} \) |
| 73 | \( 1 + 38774 T + p^{5} T^{2} \) |
| 79 | \( 1 - 33360 T + p^{5} T^{2} \) |
| 83 | \( 1 - 16716 T + p^{5} T^{2} \) |
| 89 | \( 1 - 101370 T + p^{5} T^{2} \) |
| 97 | \( 1 + 119038 T + p^{5} T^{2} \) |
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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
−10.80190430644786197843231656721, −9.876535428533976510117621177181, −8.983025923280385841352666233353, −7.962636256946231908485602253641, −6.75817530437946425666347344241, −6.04759242787700065344314799339, −4.78919962335594940297119523027, −3.45901557683531458464719008344, −2.49611108850417448478443315898, −0.34384914638445958329004326917,
0.34384914638445958329004326917, 2.49611108850417448478443315898, 3.45901557683531458464719008344, 4.78919962335594940297119523027, 6.04759242787700065344314799339, 6.75817530437946425666347344241, 7.962636256946231908485602253641, 8.983025923280385841352666233353, 9.876535428533976510117621177181, 10.80190430644786197843231656721
