| L(s) = 1 | − 2-s + 4-s − 4.24·5-s − 7-s − 8-s + 4.24·10-s + 1.05·11-s + 0.941·13-s + 14-s + 16-s − 5.30·17-s + 5.55·19-s − 4.24·20-s − 1.05·22-s + 23-s + 13.0·25-s − 0.941·26-s − 28-s + 9.43·29-s − 3.30·31-s − 32-s + 5.30·34-s + 4.24·35-s − 8.61·37-s − 5.55·38-s + 4.24·40-s + 9.11·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s − 1.90·5-s − 0.377·7-s − 0.353·8-s + 1.34·10-s + 0.319·11-s + 0.261·13-s + 0.267·14-s + 0.250·16-s − 1.28·17-s + 1.27·19-s − 0.950·20-s − 0.225·22-s + 0.208·23-s + 2.61·25-s − 0.184·26-s − 0.188·28-s + 1.75·29-s − 0.594·31-s − 0.176·32-s + 0.910·34-s + 0.718·35-s − 1.41·37-s − 0.901·38-s + 0.671·40-s + 1.42·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2898 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2898 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| good | 5 | \( 1 + 4.24T + 5T^{2} \) |
| 11 | \( 1 - 1.05T + 11T^{2} \) |
| 13 | \( 1 - 0.941T + 13T^{2} \) |
| 17 | \( 1 + 5.30T + 17T^{2} \) |
| 19 | \( 1 - 5.55T + 19T^{2} \) |
| 29 | \( 1 - 9.43T + 29T^{2} \) |
| 31 | \( 1 + 3.30T + 31T^{2} \) |
| 37 | \( 1 + 8.61T + 37T^{2} \) |
| 41 | \( 1 - 9.11T + 41T^{2} \) |
| 43 | \( 1 + 1.88T + 43T^{2} \) |
| 47 | \( 1 + 3.30T + 47T^{2} \) |
| 53 | \( 1 + 4.61T + 53T^{2} \) |
| 59 | \( 1 - 6.74T + 59T^{2} \) |
| 61 | \( 1 + 6.13T + 61T^{2} \) |
| 67 | \( 1 - 1.05T + 67T^{2} \) |
| 71 | \( 1 - 14.6T + 71T^{2} \) |
| 73 | \( 1 + 6.73T + 73T^{2} \) |
| 79 | \( 1 - 10.5T + 79T^{2} \) |
| 83 | \( 1 + 9.05T + 83T^{2} \) |
| 89 | \( 1 + 3.19T + 89T^{2} \) |
| 97 | \( 1 + 3.68T + 97T^{2} \) |
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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
−8.381068959479186751698717038347, −7.75004907241360910552687638188, −6.96154654186432160795905137037, −6.56649988519475672613029823752, −5.20014579249968069907809259193, −4.28628418224061057788759387145, −3.52774778777628903102910153411, −2.73535134715169680151016966659, −1.11164277974465395708110476555, 0,
1.11164277974465395708110476555, 2.73535134715169680151016966659, 3.52774778777628903102910153411, 4.28628418224061057788759387145, 5.20014579249968069907809259193, 6.56649988519475672613029823752, 6.96154654186432160795905137037, 7.75004907241360910552687638188, 8.381068959479186751698717038347
