L(s) = 1 | − 2.44·3-s + 1.41·5-s + 2.99·9-s − 11-s + 1.03·13-s − 3.46·15-s + 1.03·17-s − 0.535·23-s − 2.99·25-s + 3.46·29-s − 5.27·31-s + 2.44·33-s + 2·37-s − 2.53·39-s + 6.69·41-s − 5.46·43-s + 4.24·45-s + 0.378·47-s − 2.53·51-s − 6.92·53-s − 1.41·55-s − 8.10·59-s + 3.86·61-s + 1.46·65-s − 4.53·67-s + 1.31·69-s − 6.39·71-s + ⋯ |
L(s) = 1 | − 1.41·3-s + 0.632·5-s + 0.999·9-s − 0.301·11-s + 0.287·13-s − 0.894·15-s + 0.251·17-s − 0.111·23-s − 0.599·25-s + 0.643·29-s − 0.947·31-s + 0.426·33-s + 0.328·37-s − 0.406·39-s + 1.04·41-s − 0.833·43-s + 0.632·45-s + 0.0552·47-s − 0.355·51-s − 0.951·53-s − 0.190·55-s − 1.05·59-s + 0.494·61-s + 0.181·65-s − 0.554·67-s + 0.158·69-s − 0.758·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8624 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
good | 3 | \( 1 + 2.44T + 3T^{2} \) |
| 5 | \( 1 - 1.41T + 5T^{2} \) |
| 13 | \( 1 - 1.03T + 13T^{2} \) |
| 17 | \( 1 - 1.03T + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + 0.535T + 23T^{2} \) |
| 29 | \( 1 - 3.46T + 29T^{2} \) |
| 31 | \( 1 + 5.27T + 31T^{2} \) |
| 37 | \( 1 - 2T + 37T^{2} \) |
| 41 | \( 1 - 6.69T + 41T^{2} \) |
| 43 | \( 1 + 5.46T + 43T^{2} \) |
| 47 | \( 1 - 0.378T + 47T^{2} \) |
| 53 | \( 1 + 6.92T + 53T^{2} \) |
| 59 | \( 1 + 8.10T + 59T^{2} \) |
| 61 | \( 1 - 3.86T + 61T^{2} \) |
| 67 | \( 1 + 4.53T + 67T^{2} \) |
| 71 | \( 1 + 6.39T + 71T^{2} \) |
| 73 | \( 1 - 9.52T + 73T^{2} \) |
| 79 | \( 1 - 8.39T + 79T^{2} \) |
| 83 | \( 1 - 2.07T + 83T^{2} \) |
| 89 | \( 1 + 9.89T + 89T^{2} \) |
| 97 | \( 1 - 0.656T + 97T^{2} \) |
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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
−7.31513097620195005393581530781, −6.42697724467303570728744638468, −6.08423988944490430120387085914, −5.42657063920125394155661875507, −4.88357449934237698704562413057, −4.06306825365886045405485368723, −3.06243826288617860731427623447, −2.01736309412868891379033210669, −1.09991191362984260549444044840, 0,
1.09991191362984260549444044840, 2.01736309412868891379033210669, 3.06243826288617860731427623447, 4.06306825365886045405485368723, 4.88357449934237698704562413057, 5.42657063920125394155661875507, 6.08423988944490430120387085914, 6.42697724467303570728744638468, 7.31513097620195005393581530781