| L(s) = 1 | − 4·2-s + 16·4-s + 45·5-s − 121·7-s − 64·8-s − 180·10-s + 381·11-s − 100·13-s + 484·14-s + 256·16-s − 933·17-s + 361·19-s + 720·20-s − 1.52e3·22-s + 552·23-s − 1.10e3·25-s + 400·26-s − 1.93e3·28-s − 2.39e3·29-s − 4.02e3·31-s − 1.02e3·32-s + 3.73e3·34-s − 5.44e3·35-s + 9.18e3·37-s − 1.44e3·38-s − 2.88e3·40-s + 2.25e3·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.804·5-s − 0.933·7-s − 0.353·8-s − 0.569·10-s + 0.949·11-s − 0.164·13-s + 0.659·14-s + 1/4·16-s − 0.782·17-s + 0.229·19-s + 0.402·20-s − 0.671·22-s + 0.217·23-s − 0.351·25-s + 0.116·26-s − 0.466·28-s − 0.528·29-s − 0.752·31-s − 0.176·32-s + 0.553·34-s − 0.751·35-s + 1.10·37-s − 0.162·38-s − 0.284·40-s + 0.209·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p^{2} T \) |
| 3 | \( 1 \) |
| 19 | \( 1 - p^{2} T \) |
| good | 5 | \( 1 - 9 p T + p^{5} T^{2} \) |
| 7 | \( 1 + 121 T + p^{5} T^{2} \) |
| 11 | \( 1 - 381 T + p^{5} T^{2} \) |
| 13 | \( 1 + 100 T + p^{5} T^{2} \) |
| 17 | \( 1 + 933 T + p^{5} T^{2} \) |
| 23 | \( 1 - 24 p T + p^{5} T^{2} \) |
| 29 | \( 1 + 2394 T + p^{5} T^{2} \) |
| 31 | \( 1 + 4024 T + p^{5} T^{2} \) |
| 37 | \( 1 - 9182 T + p^{5} T^{2} \) |
| 41 | \( 1 - 2250 T + p^{5} T^{2} \) |
| 43 | \( 1 + 23377 T + p^{5} T^{2} \) |
| 47 | \( 1 - 26595 T + p^{5} T^{2} \) |
| 53 | \( 1 - 16008 T + p^{5} T^{2} \) |
| 59 | \( 1 - 126 T + p^{5} T^{2} \) |
| 61 | \( 1 - 21335 T + p^{5} T^{2} \) |
| 67 | \( 1 + 51760 T + p^{5} T^{2} \) |
| 71 | \( 1 + 8574 T + p^{5} T^{2} \) |
| 73 | \( 1 - 11153 T + p^{5} T^{2} \) |
| 79 | \( 1 + 1660 T + p^{5} T^{2} \) |
| 83 | \( 1 + 95964 T + p^{5} T^{2} \) |
| 89 | \( 1 + 118848 T + p^{5} T^{2} \) |
| 97 | \( 1 + 153760 T + p^{5} T^{2} \) |
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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
−9.924394462201038354584369120938, −9.432918467656973818081442048768, −8.641093830063899674420986043323, −7.24918345816918428540666641487, −6.47660712111092339913612552979, −5.61434282693182433176604177643, −3.96716642398053183327732009903, −2.63942573510925293405937760198, −1.45271868559632247634803097459, 0,
1.45271868559632247634803097459, 2.63942573510925293405937760198, 3.96716642398053183327732009903, 5.61434282693182433176604177643, 6.47660712111092339913612552979, 7.24918345816918428540666641487, 8.641093830063899674420986043323, 9.432918467656973818081442048768, 9.924394462201038354584369120938
