| L(s) = 1 | − 0.307·2-s − 3-s − 1.90·4-s − 1.88·5-s + 0.307·6-s − 3.18·7-s + 1.20·8-s + 9-s + 0.578·10-s − 3.08·11-s + 1.90·12-s − 13-s + 0.979·14-s + 1.88·15-s + 3.44·16-s − 4.84·17-s − 0.307·18-s + 5.64·19-s + 3.58·20-s + 3.18·21-s + 0.950·22-s + 6.40·23-s − 1.20·24-s − 1.46·25-s + 0.307·26-s − 27-s + 6.06·28-s + ⋯ |
| L(s) = 1 | − 0.217·2-s − 0.577·3-s − 0.952·4-s − 0.841·5-s + 0.125·6-s − 1.20·7-s + 0.424·8-s + 0.333·9-s + 0.182·10-s − 0.931·11-s + 0.550·12-s − 0.277·13-s + 0.261·14-s + 0.485·15-s + 0.860·16-s − 1.17·17-s − 0.0725·18-s + 1.29·19-s + 0.801·20-s + 0.694·21-s + 0.202·22-s + 1.33·23-s − 0.245·24-s − 0.292·25-s + 0.0603·26-s − 0.192·27-s + 1.14·28-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 | \( 1 + 0.307T + 2T^{2} \) |
| 5 | \( 1 + 1.88T + 5T^{2} \) |
| 7 | \( 1 + 3.18T + 7T^{2} \) |
| 11 | \( 1 + 3.08T + 11T^{2} \) |
| 17 | \( 1 + 4.84T + 17T^{2} \) |
| 19 | \( 1 - 5.64T + 19T^{2} \) |
| 23 | \( 1 - 6.40T + 23T^{2} \) |
| 29 | \( 1 - 1.13T + 29T^{2} \) |
| 31 | \( 1 - 1.55T + 31T^{2} \) |
| 37 | \( 1 - 9.63T + 37T^{2} \) |
| 41 | \( 1 + 2.54T + 41T^{2} \) |
| 43 | \( 1 + 4.37T + 43T^{2} \) |
| 47 | \( 1 - 9.16T + 47T^{2} \) |
| 53 | \( 1 - 2.80T + 53T^{2} \) |
| 59 | \( 1 + 3.55T + 59T^{2} \) |
| 61 | \( 1 - 8.59T + 61T^{2} \) |
| 67 | \( 1 + 10.8T + 67T^{2} \) |
| 71 | \( 1 - 1.88T + 71T^{2} \) |
| 73 | \( 1 - 12.0T + 73T^{2} \) |
| 79 | \( 1 - 7.69T + 79T^{2} \) |
| 83 | \( 1 - 0.322T + 83T^{2} \) |
| 89 | \( 1 + 11.6T + 89T^{2} \) |
| 97 | \( 1 + 11.5T + 97T^{2} \) |
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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
−8.041064626727599357194298305857, −7.38576117392028157658952611802, −6.72854262302826479927865009926, −5.74904265831298739953645772464, −5.02730119887571651945980977728, −4.34465616320904177220118546649, −3.50176684279755864678365414262, −2.66914392111120472195959386108, −0.853871481071792216944875332681, 0,
0.853871481071792216944875332681, 2.66914392111120472195959386108, 3.50176684279755864678365414262, 4.34465616320904177220118546649, 5.02730119887571651945980977728, 5.74904265831298739953645772464, 6.72854262302826479927865009926, 7.38576117392028157658952611802, 8.041064626727599357194298305857
