| L(s) = 1 | − 1.05·2-s − 2.79·3-s − 0.888·4-s + 2.94·6-s + 1.63·7-s + 3.04·8-s + 4.80·9-s + 3.48·11-s + 2.48·12-s − 1.54·13-s − 1.72·14-s − 1.43·16-s − 5.66·17-s − 5.06·18-s + 8.06·19-s − 4.57·21-s − 3.67·22-s + 9.08·23-s − 8.50·24-s + 1.62·26-s − 5.04·27-s − 1.45·28-s − 5.83·29-s − 0.515·31-s − 4.57·32-s − 9.72·33-s + 5.97·34-s + ⋯ |
| L(s) = 1 | − 0.745·2-s − 1.61·3-s − 0.444·4-s + 1.20·6-s + 0.618·7-s + 1.07·8-s + 1.60·9-s + 1.04·11-s + 0.716·12-s − 0.427·13-s − 0.461·14-s − 0.358·16-s − 1.37·17-s − 1.19·18-s + 1.84·19-s − 0.998·21-s − 0.782·22-s + 1.89·23-s − 1.73·24-s + 0.318·26-s − 0.971·27-s − 0.274·28-s − 1.08·29-s − 0.0925·31-s − 0.809·32-s − 1.69·33-s + 1.02·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| 241 | \( 1 + T \) |
| good | 2 | \( 1 + 1.05T + 2T^{2} \) |
| 3 | \( 1 + 2.79T + 3T^{2} \) |
| 7 | \( 1 - 1.63T + 7T^{2} \) |
| 11 | \( 1 - 3.48T + 11T^{2} \) |
| 13 | \( 1 + 1.54T + 13T^{2} \) |
| 17 | \( 1 + 5.66T + 17T^{2} \) |
| 19 | \( 1 - 8.06T + 19T^{2} \) |
| 23 | \( 1 - 9.08T + 23T^{2} \) |
| 29 | \( 1 + 5.83T + 29T^{2} \) |
| 31 | \( 1 + 0.515T + 31T^{2} \) |
| 37 | \( 1 + 8.17T + 37T^{2} \) |
| 41 | \( 1 + 2.50T + 41T^{2} \) |
| 43 | \( 1 - 2.34T + 43T^{2} \) |
| 47 | \( 1 + 7.36T + 47T^{2} \) |
| 53 | \( 1 - 6.12T + 53T^{2} \) |
| 59 | \( 1 - 1.35T + 59T^{2} \) |
| 61 | \( 1 + 11.6T + 61T^{2} \) |
| 67 | \( 1 + 1.12T + 67T^{2} \) |
| 71 | \( 1 + 6.10T + 71T^{2} \) |
| 73 | \( 1 + 5.57T + 73T^{2} \) |
| 79 | \( 1 + 9.29T + 79T^{2} \) |
| 83 | \( 1 - 4.55T + 83T^{2} \) |
| 89 | \( 1 + 2.59T + 89T^{2} \) |
| 97 | \( 1 - 12.3T + 97T^{2} \) |
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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
−7.39423782952175713431528475133, −7.21343377604964576432237451095, −6.38415800472500481526526613145, −5.40791371577441446755061557449, −4.94249444637394580471317135769, −4.43038608696478086649722249358, −3.36538620470568378174874032805, −1.69080700863922103518102183862, −1.06264782617351503490330471269, 0,
1.06264782617351503490330471269, 1.69080700863922103518102183862, 3.36538620470568378174874032805, 4.43038608696478086649722249358, 4.94249444637394580471317135769, 5.40791371577441446755061557449, 6.38415800472500481526526613145, 7.21343377604964576432237451095, 7.39423782952175713431528475133
