| L(s) = 1 | − 2-s + 4-s − 3.53·5-s − 4.75·7-s − 8-s + 3.53·10-s − 3.18·11-s − 1.06·13-s + 4.75·14-s + 16-s + 5.75·19-s − 3.53·20-s + 3.18·22-s + 0.610·23-s + 7.47·25-s + 1.06·26-s − 4.75·28-s + 2.02·29-s + 8.92·31-s − 32-s + 16.8·35-s − 5.30·37-s − 5.75·38-s + 3.53·40-s + 6.45·41-s + 6·43-s − 3.18·44-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s − 1.57·5-s − 1.79·7-s − 0.353·8-s + 1.11·10-s − 0.960·11-s − 0.295·13-s + 1.27·14-s + 0.250·16-s + 1.32·19-s − 0.789·20-s + 0.679·22-s + 0.127·23-s + 1.49·25-s + 0.208·26-s − 0.899·28-s + 0.375·29-s + 1.60·31-s − 0.176·32-s + 2.84·35-s − 0.872·37-s − 0.934·38-s + 0.558·40-s + 1.00·41-s + 0.914·43-s − 0.480·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5202 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5202 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 17 | \( 1 \) |
| good | 5 | \( 1 + 3.53T + 5T^{2} \) |
| 7 | \( 1 + 4.75T + 7T^{2} \) |
| 11 | \( 1 + 3.18T + 11T^{2} \) |
| 13 | \( 1 + 1.06T + 13T^{2} \) |
| 19 | \( 1 - 5.75T + 19T^{2} \) |
| 23 | \( 1 - 0.610T + 23T^{2} \) |
| 29 | \( 1 - 2.02T + 29T^{2} \) |
| 31 | \( 1 - 8.92T + 31T^{2} \) |
| 37 | \( 1 + 5.30T + 37T^{2} \) |
| 41 | \( 1 - 6.45T + 41T^{2} \) |
| 43 | \( 1 - 6T + 43T^{2} \) |
| 47 | \( 1 + 13.1T + 47T^{2} \) |
| 53 | \( 1 - 1.79T + 53T^{2} \) |
| 59 | \( 1 - 9.04T + 59T^{2} \) |
| 61 | \( 1 + 1.01T + 61T^{2} \) |
| 67 | \( 1 + 2.58T + 67T^{2} \) |
| 71 | \( 1 - 10.4T + 71T^{2} \) |
| 73 | \( 1 + 13.9T + 73T^{2} \) |
| 79 | \( 1 + 4.73T + 79T^{2} \) |
| 83 | \( 1 + 0.0641T + 83T^{2} \) |
| 89 | \( 1 + 5.67T + 89T^{2} \) |
| 97 | \( 1 - 3.02T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.84543758743832314593298908399, −7.23768882949819347895966505219, −6.72306948123593557723604485813, −5.84567881484033796010209949072, −4.88356540015671209240519104996, −3.89331937747355494066192900313, −3.11422889228566283568947802026, −2.70419781495002138424063964544, −0.835244565750533502984906921141, 0,
0.835244565750533502984906921141, 2.70419781495002138424063964544, 3.11422889228566283568947802026, 3.89331937747355494066192900313, 4.88356540015671209240519104996, 5.84567881484033796010209949072, 6.72306948123593557723604485813, 7.23768882949819347895966505219, 7.84543758743832314593298908399
