| L(s) = 1 | − 14·3-s + 115·9-s + 46·11-s + 574·17-s − 434·19-s − 476·27-s − 644·33-s − 1.24e3·41-s − 3.50e3·43-s + 2.40e3·49-s − 8.03e3·51-s + 6.07e3·57-s + 238·59-s − 5.13e3·67-s − 9.50e3·73-s − 2.65e3·81-s + 1.11e4·83-s + 5.47e3·89-s + 9.98e3·97-s + 5.29e3·99-s + 8.78e3·107-s + 1.59e4·113-s + ⋯ |
| L(s) = 1 | − 1.55·3-s + 1.41·9-s + 0.380·11-s + 1.98·17-s − 1.20·19-s − 0.652·27-s − 0.591·33-s − 0.741·41-s − 1.89·43-s + 49-s − 3.08·51-s + 1.87·57-s + 0.0683·59-s − 1.14·67-s − 1.78·73-s − 0.404·81-s + 1.62·83-s + 0.691·89-s + 1.06·97-s + 0.539·99-s + 0.767·107-s + 1.24·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.022915231\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.022915231\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + 14 T + p^{4} T^{2} \) |
| 7 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 11 | \( 1 - 46 T + p^{4} T^{2} \) |
| 13 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 17 | \( 1 - 574 T + p^{4} T^{2} \) |
| 19 | \( 1 + 434 T + p^{4} T^{2} \) |
| 23 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 29 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 31 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 37 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 41 | \( 1 + 1246 T + p^{4} T^{2} \) |
| 43 | \( 1 + 3502 T + p^{4} T^{2} \) |
| 47 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 53 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 59 | \( 1 - 238 T + p^{4} T^{2} \) |
| 61 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 67 | \( 1 + 5134 T + p^{4} T^{2} \) |
| 71 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 73 | \( 1 + 9506 T + p^{4} T^{2} \) |
| 79 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 83 | \( 1 - 11186 T + p^{4} T^{2} \) |
| 89 | \( 1 - 5474 T + p^{4} T^{2} \) |
| 97 | \( 1 - 9982 T + p^{4} T^{2} \) |
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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
−10.15324570781044639211338213210, −8.896884680951272581371870219919, −7.83301405358806686711391689012, −6.85797073058456705289009824520, −6.09249964182129470275123831175, −5.38435279154819569506361767383, −4.50685588459518569431137840901, −3.35362057347561861583710488305, −1.63681407142536895138398522689, −0.56918584784163899871659417931,
0.56918584784163899871659417931, 1.63681407142536895138398522689, 3.35362057347561861583710488305, 4.50685588459518569431137840901, 5.38435279154819569506361767383, 6.09249964182129470275123831175, 6.85797073058456705289009824520, 7.83301405358806686711391689012, 8.896884680951272581371870219919, 10.15324570781044639211338213210
