| L(s) = 1 | + 4·2-s − 9.60·3-s + 16·4-s − 38.4·6-s − 16.6·7-s + 64·8-s − 150.·9-s − 693.·11-s − 153.·12-s − 169·13-s − 66.7·14-s + 256·16-s − 826.·17-s − 602.·18-s − 1.78e3·19-s + 160.·21-s − 2.77e3·22-s + 2.37e3·23-s − 614.·24-s − 676·26-s + 3.78e3·27-s − 266.·28-s + 2.02e3·29-s + 6.72e3·31-s + 1.02e3·32-s + 6.66e3·33-s − 3.30e3·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.616·3-s + 0.5·4-s − 0.435·6-s − 0.128·7-s + 0.353·8-s − 0.620·9-s − 1.72·11-s − 0.308·12-s − 0.277·13-s − 0.0909·14-s + 0.250·16-s − 0.693·17-s − 0.438·18-s − 1.13·19-s + 0.0793·21-s − 1.22·22-s + 0.936·23-s − 0.217·24-s − 0.196·26-s + 0.998·27-s − 0.0643·28-s + 0.446·29-s + 1.25·31-s + 0.176·32-s + 1.06·33-s − 0.490·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.496208418\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.496208418\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 4T \) |
| 5 | \( 1 \) |
| 13 | \( 1 + 169T \) |
| good | 3 | \( 1 + 9.60T + 243T^{2} \) |
| 7 | \( 1 + 16.6T + 1.68e4T^{2} \) |
| 11 | \( 1 + 693.T + 1.61e5T^{2} \) |
| 17 | \( 1 + 826.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 1.78e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 2.37e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 2.02e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 6.72e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 2.85e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.93e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.40e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.99e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 3.91e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 2.40e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 3.45e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 2.04e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 4.39e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 1.82e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 4.13e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 5.83e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 4.87e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.51e5T + 8.58e9T^{2} \) |
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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
−10.17728794755868556132008838544, −8.770991009791435438928106530515, −7.968868778022150678404246625404, −6.85741662499994260570537028487, −6.06721758565053115164935135011, −5.15976472861249016692667227020, −4.54005019225721683972820529774, −3.02730064186136903564756910865, −2.29995135908343013614979313403, −0.50783733117930365376203155698,
0.50783733117930365376203155698, 2.29995135908343013614979313403, 3.02730064186136903564756910865, 4.54005019225721683972820529774, 5.15976472861249016692667227020, 6.06721758565053115164935135011, 6.85741662499994260570537028487, 7.968868778022150678404246625404, 8.770991009791435438928106530515, 10.17728794755868556132008838544
