L(s) = 1 | + 2.67·2-s + 0.199·3-s + 5.16·4-s + 0.533·6-s − 2.87·7-s + 8.45·8-s − 2.96·9-s − 0.806·11-s + 1.02·12-s − 5.09·13-s − 7.69·14-s + 12.3·16-s − 0.390·17-s − 7.92·18-s + 0.489·19-s − 0.572·21-s − 2.15·22-s − 1.83·23-s + 1.68·24-s − 13.6·26-s − 1.18·27-s − 14.8·28-s − 8.05·29-s − 5.55·31-s + 16.0·32-s − 0.160·33-s − 1.04·34-s + ⋯ |
L(s) = 1 | + 1.89·2-s + 0.115·3-s + 2.58·4-s + 0.217·6-s − 1.08·7-s + 2.99·8-s − 0.986·9-s − 0.243·11-s + 0.296·12-s − 1.41·13-s − 2.05·14-s + 3.07·16-s − 0.0947·17-s − 1.86·18-s + 0.112·19-s − 0.124·21-s − 0.460·22-s − 0.383·23-s + 0.344·24-s − 2.67·26-s − 0.228·27-s − 2.80·28-s − 1.49·29-s − 0.997·31-s + 2.83·32-s − 0.0279·33-s − 0.179·34-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 | \( 1 - 2.67T + 2T^{2} \) |
| 3 | \( 1 - 0.199T + 3T^{2} \) |
| 7 | \( 1 + 2.87T + 7T^{2} \) |
| 11 | \( 1 + 0.806T + 11T^{2} \) |
| 13 | \( 1 + 5.09T + 13T^{2} \) |
| 17 | \( 1 + 0.390T + 17T^{2} \) |
| 19 | \( 1 - 0.489T + 19T^{2} \) |
| 23 | \( 1 + 1.83T + 23T^{2} \) |
| 29 | \( 1 + 8.05T + 29T^{2} \) |
| 31 | \( 1 + 5.55T + 31T^{2} \) |
| 37 | \( 1 + 2.20T + 37T^{2} \) |
| 41 | \( 1 + 3.59T + 41T^{2} \) |
| 43 | \( 1 + 9.27T + 43T^{2} \) |
| 47 | \( 1 - 10.9T + 47T^{2} \) |
| 53 | \( 1 + 1.68T + 53T^{2} \) |
| 59 | \( 1 + 5.02T + 59T^{2} \) |
| 61 | \( 1 - 3.39T + 61T^{2} \) |
| 67 | \( 1 - 12.9T + 67T^{2} \) |
| 71 | \( 1 - 8.48T + 71T^{2} \) |
| 73 | \( 1 - 13.1T + 73T^{2} \) |
| 79 | \( 1 - 8.08T + 79T^{2} \) |
| 83 | \( 1 - 0.366T + 83T^{2} \) |
| 89 | \( 1 + 2.24T + 89T^{2} \) |
| 97 | \( 1 - 7.32T + 97T^{2} \) |
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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
−7.37446060163180351608668319149, −6.77644066538834918358525444997, −6.13692761763480607920839380105, −5.31222324013491319302494442506, −5.09243672737477054437374433884, −3.84036144215067812140258927169, −3.45387758932755018496684088453, −2.59066862003229863126347010524, −2.05246273075899837702819128316, 0,
2.05246273075899837702819128316, 2.59066862003229863126347010524, 3.45387758932755018496684088453, 3.84036144215067812140258927169, 5.09243672737477054437374433884, 5.31222324013491319302494442506, 6.13692761763480607920839380105, 6.77644066538834918358525444997, 7.37446060163180351608668319149