| L(s) = 1 | − 1.82·2-s − 3-s + 1.34·4-s + 4.14·5-s + 1.82·6-s − 1.02·7-s + 1.19·8-s + 9-s − 7.57·10-s + 1.59·11-s − 1.34·12-s + 13-s + 1.86·14-s − 4.14·15-s − 4.88·16-s + 3.45·17-s − 1.82·18-s − 7.53·19-s + 5.57·20-s + 1.02·21-s − 2.92·22-s + 1.69·23-s − 1.19·24-s + 12.1·25-s − 1.82·26-s − 27-s − 1.37·28-s + ⋯ |
| L(s) = 1 | − 1.29·2-s − 0.577·3-s + 0.672·4-s + 1.85·5-s + 0.746·6-s − 0.386·7-s + 0.423·8-s + 0.333·9-s − 2.39·10-s + 0.481·11-s − 0.388·12-s + 0.277·13-s + 0.499·14-s − 1.06·15-s − 1.22·16-s + 0.838·17-s − 0.431·18-s − 1.72·19-s + 1.24·20-s + 0.222·21-s − 0.622·22-s + 0.354·23-s − 0.244·24-s + 2.43·25-s − 0.358·26-s − 0.192·27-s − 0.259·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 103 | \( 1 - T \) |
| good | 2 | \( 1 + 1.82T + 2T^{2} \) |
| 5 | \( 1 - 4.14T + 5T^{2} \) |
| 7 | \( 1 + 1.02T + 7T^{2} \) |
| 11 | \( 1 - 1.59T + 11T^{2} \) |
| 17 | \( 1 - 3.45T + 17T^{2} \) |
| 19 | \( 1 + 7.53T + 19T^{2} \) |
| 23 | \( 1 - 1.69T + 23T^{2} \) |
| 29 | \( 1 + 2.96T + 29T^{2} \) |
| 31 | \( 1 + 4.28T + 31T^{2} \) |
| 37 | \( 1 + 11.2T + 37T^{2} \) |
| 41 | \( 1 - 4.76T + 41T^{2} \) |
| 43 | \( 1 + 8.21T + 43T^{2} \) |
| 47 | \( 1 + 7.29T + 47T^{2} \) |
| 53 | \( 1 - 2.79T + 53T^{2} \) |
| 59 | \( 1 + 10.5T + 59T^{2} \) |
| 61 | \( 1 + 13.2T + 61T^{2} \) |
| 67 | \( 1 + 0.576T + 67T^{2} \) |
| 71 | \( 1 + 9.51T + 71T^{2} \) |
| 73 | \( 1 + 3.46T + 73T^{2} \) |
| 79 | \( 1 + 0.683T + 79T^{2} \) |
| 83 | \( 1 + 16.7T + 83T^{2} \) |
| 89 | \( 1 - 0.765T + 89T^{2} \) |
| 97 | \( 1 - 5.58T + 97T^{2} \) |
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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
−8.384487948199743012454796190024, −7.27617301251108355558632758720, −6.59928969613737610629346794862, −6.09060243062387363193308380540, −5.32224708219283423491551211753, −4.45257255630200735831197450299, −3.13408262656792471032885625284, −1.80607798746717378315729003615, −1.50651206610200529526508413916, 0,
1.50651206610200529526508413916, 1.80607798746717378315729003615, 3.13408262656792471032885625284, 4.45257255630200735831197450299, 5.32224708219283423491551211753, 6.09060243062387363193308380540, 6.59928969613737610629346794862, 7.27617301251108355558632758720, 8.384487948199743012454796190024
