| L(s) = 1 | − 2-s + 4-s − 5-s − 7-s − 8-s − 3·9-s + 10-s − 4·11-s − 6·13-s + 14-s + 16-s − 2·17-s + 3·18-s − 20-s + 4·22-s + 25-s + 6·26-s − 28-s − 8·31-s − 32-s + 2·34-s + 35-s − 3·36-s + 10·37-s + 40-s − 2·41-s − 4·43-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.377·7-s − 0.353·8-s − 9-s + 0.316·10-s − 1.20·11-s − 1.66·13-s + 0.267·14-s + 1/4·16-s − 0.485·17-s + 0.707·18-s − 0.223·20-s + 0.852·22-s + 1/5·25-s + 1.17·26-s − 0.188·28-s − 1.43·31-s − 0.176·32-s + 0.342·34-s + 0.169·35-s − 1/2·36-s + 1.64·37-s + 0.158·40-s − 0.312·41-s − 0.609·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 58870 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 58870 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 29 | \( 1 \) |
| good | 3 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
| 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
−14.72527885456749, −14.33828460037114, −13.33721077509525, −13.12021473768286, −12.48030742725488, −11.95830941688678, −11.49426442993914, −10.95841578279550, −10.54785719736099, −9.786293210852000, −9.620401319323627, −8.828646387031657, −8.426933222350932, −7.760805350338122, −7.445521207037271, −6.939599813441571, −6.123625868767619, −5.682894012910577, −4.925640873804950, −4.579444484692473, −3.481746507796177, −2.970785018446178, −2.441649481142219, −1.846405371604908, −0.5076631430696542, 0,
0.5076631430696542, 1.846405371604908, 2.441649481142219, 2.970785018446178, 3.481746507796177, 4.579444484692473, 4.925640873804950, 5.682894012910577, 6.123625868767619, 6.939599813441571, 7.445521207037271, 7.760805350338122, 8.426933222350932, 8.828646387031657, 9.620401319323627, 9.786293210852000, 10.54785719736099, 10.95841578279550, 11.49426442993914, 11.95830941688678, 12.48030742725488, 13.12021473768286, 13.33721077509525, 14.33828460037114, 14.72527885456749
