| L(s) = 1 | + 2-s − 3-s + 4-s − 6-s + 8-s + 9-s − 2.58·11-s − 12-s + 2.82·13-s + 16-s + 1.17·17-s + 18-s + 19-s − 2.58·22-s − 4.24·23-s − 24-s − 5·25-s + 2.82·26-s − 27-s − 8·29-s − 3.65·31-s + 32-s + 2.58·33-s + 1.17·34-s + 36-s − 2·37-s + 38-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.408·6-s + 0.353·8-s + 0.333·9-s − 0.779·11-s − 0.288·12-s + 0.784·13-s + 0.250·16-s + 0.284·17-s + 0.235·18-s + 0.229·19-s − 0.551·22-s − 0.884·23-s − 0.204·24-s − 25-s + 0.554·26-s − 0.192·27-s − 1.48·29-s − 0.656·31-s + 0.176·32-s + 0.450·33-s + 0.200·34-s + 0.166·36-s − 0.328·37-s + 0.162·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5586 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5586 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 5 | \( 1 + 5T^{2} \) |
| 11 | \( 1 + 2.58T + 11T^{2} \) |
| 13 | \( 1 - 2.82T + 13T^{2} \) |
| 17 | \( 1 - 1.17T + 17T^{2} \) |
| 23 | \( 1 + 4.24T + 23T^{2} \) |
| 29 | \( 1 + 8T + 29T^{2} \) |
| 31 | \( 1 + 3.65T + 31T^{2} \) |
| 37 | \( 1 + 2T + 37T^{2} \) |
| 41 | \( 1 - 2.58T + 41T^{2} \) |
| 43 | \( 1 + 6.82T + 43T^{2} \) |
| 47 | \( 1 - 7.65T + 47T^{2} \) |
| 53 | \( 1 + 2T + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 2.58T + 61T^{2} \) |
| 67 | \( 1 + 4.24T + 67T^{2} \) |
| 71 | \( 1 - 14.1T + 71T^{2} \) |
| 73 | \( 1 + 9.41T + 73T^{2} \) |
| 79 | \( 1 - 5.89T + 79T^{2} \) |
| 83 | \( 1 - 9.65T + 83T^{2} \) |
| 89 | \( 1 + 2.58T + 89T^{2} \) |
| 97 | \( 1 + 14.8T + 97T^{2} \) |
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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
−7.67933461268933182030600759577, −6.99973521714588830156063450477, −6.07192928588972586626584586113, −5.65592042630604715287241778313, −5.02932569178797587271488034354, −4.01112890684963712767620301358, −3.53458276785813867974005841305, −2.36814324657295190567216902329, −1.48383647159242796469570490988, 0,
1.48383647159242796469570490988, 2.36814324657295190567216902329, 3.53458276785813867974005841305, 4.01112890684963712767620301358, 5.02932569178797587271488034354, 5.65592042630604715287241778313, 6.07192928588972586626584586113, 6.99973521714588830156063450477, 7.67933461268933182030600759577
