| L(s) = 1 | − 2·2-s − 8·3-s + 4·4-s + 14·5-s + 16·6-s − 8·8-s + 37·9-s − 28·10-s − 28·11-s − 32·12-s − 18·13-s − 112·15-s + 16·16-s − 74·17-s − 74·18-s − 80·19-s + 56·20-s + 56·22-s − 112·23-s + 64·24-s + 71·25-s + 36·26-s − 80·27-s + 190·29-s + 224·30-s − 72·31-s − 32·32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.53·3-s + 1/2·4-s + 1.25·5-s + 1.08·6-s − 0.353·8-s + 1.37·9-s − 0.885·10-s − 0.767·11-s − 0.769·12-s − 0.384·13-s − 1.92·15-s + 1/4·16-s − 1.05·17-s − 0.968·18-s − 0.965·19-s + 0.626·20-s + 0.542·22-s − 1.01·23-s + 0.544·24-s + 0.567·25-s + 0.271·26-s − 0.570·27-s + 1.21·29-s + 1.36·30-s − 0.417·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p T \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + 8 T + p^{3} T^{2} \) |
| 5 | \( 1 - 14 T + p^{3} T^{2} \) |
| 11 | \( 1 + 28 T + p^{3} T^{2} \) |
| 13 | \( 1 + 18 T + p^{3} T^{2} \) |
| 17 | \( 1 + 74 T + p^{3} T^{2} \) |
| 19 | \( 1 + 80 T + p^{3} T^{2} \) |
| 23 | \( 1 + 112 T + p^{3} T^{2} \) |
| 29 | \( 1 - 190 T + p^{3} T^{2} \) |
| 31 | \( 1 + 72 T + p^{3} T^{2} \) |
| 37 | \( 1 + 346 T + p^{3} T^{2} \) |
| 41 | \( 1 + 162 T + p^{3} T^{2} \) |
| 43 | \( 1 + 412 T + p^{3} T^{2} \) |
| 47 | \( 1 + 24 T + p^{3} T^{2} \) |
| 53 | \( 1 - 6 p T + p^{3} T^{2} \) |
| 59 | \( 1 - 200 T + p^{3} T^{2} \) |
| 61 | \( 1 - 198 T + p^{3} T^{2} \) |
| 67 | \( 1 + 716 T + p^{3} T^{2} \) |
| 71 | \( 1 - 392 T + p^{3} T^{2} \) |
| 73 | \( 1 + 538 T + p^{3} T^{2} \) |
| 79 | \( 1 - 240 T + p^{3} T^{2} \) |
| 83 | \( 1 - 1072 T + p^{3} T^{2} \) |
| 89 | \( 1 + 810 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1354 T + p^{3} T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.68367359460274087992425673239, −11.65704800172437385928584918956, −10.47030007805104726419206257293, −10.10645684573923676463380807841, −8.610106374520820032447895540792, −6.86395319809953272387554387683, −6.03163790570187630286806748029, −4.95526281107430216567908247731, −2.01926087041712851147347298871, 0,
2.01926087041712851147347298871, 4.95526281107430216567908247731, 6.03163790570187630286806748029, 6.86395319809953272387554387683, 8.610106374520820032447895540792, 10.10645684573923676463380807841, 10.47030007805104726419206257293, 11.65704800172437385928584918956, 12.68367359460274087992425673239
