| L(s) = 1 | + 2-s + 3.14·3-s + 4-s − 4.14·5-s + 3.14·6-s − 7-s + 8-s + 6.91·9-s − 4.14·10-s + 2.87·11-s + 3.14·12-s − 6.91·13-s − 14-s − 13.0·15-s + 16-s − 4.02·17-s + 6.91·18-s + 0.270·19-s − 4.14·20-s − 3.14·21-s + 2.87·22-s + 3.14·24-s + 12.2·25-s − 6.91·26-s + 12.3·27-s − 28-s − 3.02·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.81·3-s + 0.5·4-s − 1.85·5-s + 1.28·6-s − 0.377·7-s + 0.353·8-s + 2.30·9-s − 1.31·10-s + 0.867·11-s + 0.908·12-s − 1.91·13-s − 0.267·14-s − 3.37·15-s + 0.250·16-s − 0.976·17-s + 1.62·18-s + 0.0620·19-s − 0.927·20-s − 0.687·21-s + 0.613·22-s + 0.642·24-s + 2.44·25-s − 1.35·26-s + 2.37·27-s − 0.188·28-s − 0.562·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7406 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7406 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 \) |
| good | 3 | \( 1 - 3.14T + 3T^{2} \) |
| 5 | \( 1 + 4.14T + 5T^{2} \) |
| 11 | \( 1 - 2.87T + 11T^{2} \) |
| 13 | \( 1 + 6.91T + 13T^{2} \) |
| 17 | \( 1 + 4.02T + 17T^{2} \) |
| 19 | \( 1 - 0.270T + 19T^{2} \) |
| 29 | \( 1 + 3.02T + 29T^{2} \) |
| 31 | \( 1 + 2.76T + 31T^{2} \) |
| 37 | \( 1 - 0.765T + 37T^{2} \) |
| 41 | \( 1 + 1.38T + 41T^{2} \) |
| 43 | \( 1 + 10.5T + 43T^{2} \) |
| 47 | \( 1 + 6.18T + 47T^{2} \) |
| 53 | \( 1 - 1.02T + 53T^{2} \) |
| 59 | \( 1 + 8.79T + 59T^{2} \) |
| 61 | \( 1 + 2.97T + 61T^{2} \) |
| 67 | \( 1 + 3.03T + 67T^{2} \) |
| 71 | \( 1 + 4.58T + 71T^{2} \) |
| 73 | \( 1 + 0.459T + 73T^{2} \) |
| 79 | \( 1 + 7.53T + 79T^{2} \) |
| 83 | \( 1 - 1.91T + 83T^{2} \) |
| 89 | \( 1 - 8.33T + 89T^{2} \) |
| 97 | \( 1 - 13.8T + 97T^{2} \) |
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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
−7.52074919482052143217930112222, −7.11245995268606383809414226363, −6.54647671611254657973259388030, −4.91710541375815932031860087147, −4.46533939104497374468321653589, −3.80560780480601516450492076926, −3.26815446715331075518538057678, −2.63753062490435831337424537321, −1.69605703793822079265150285187, 0,
1.69605703793822079265150285187, 2.63753062490435831337424537321, 3.26815446715331075518538057678, 3.80560780480601516450492076926, 4.46533939104497374468321653589, 4.91710541375815932031860087147, 6.54647671611254657973259388030, 7.11245995268606383809414226363, 7.52074919482052143217930112222
