L(s) = 1 | + 2.06·2-s − 2.17·3-s + 2.27·4-s − 4.49·6-s − 2.39·7-s + 0.576·8-s + 1.71·9-s + 4.70·11-s − 4.94·12-s + 0.253·13-s − 4.94·14-s − 3.36·16-s − 7.11·17-s + 3.54·18-s + 7.42·19-s + 5.19·21-s + 9.72·22-s + 0.555·23-s − 1.25·24-s + 0.524·26-s + 2.79·27-s − 5.44·28-s − 0.396·29-s + 9.01·31-s − 8.11·32-s − 10.2·33-s − 14.7·34-s + ⋯ |
L(s) = 1 | + 1.46·2-s − 1.25·3-s + 1.13·4-s − 1.83·6-s − 0.903·7-s + 0.203·8-s + 0.571·9-s + 1.41·11-s − 1.42·12-s + 0.0703·13-s − 1.32·14-s − 0.841·16-s − 1.72·17-s + 0.836·18-s + 1.70·19-s + 1.13·21-s + 2.07·22-s + 0.115·23-s − 0.255·24-s + 0.102·26-s + 0.536·27-s − 1.02·28-s − 0.0735·29-s + 1.61·31-s − 1.43·32-s − 1.77·33-s − 2.52·34-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 | \( 1 - 2.06T + 2T^{2} \) |
| 3 | \( 1 + 2.17T + 3T^{2} \) |
| 7 | \( 1 + 2.39T + 7T^{2} \) |
| 11 | \( 1 - 4.70T + 11T^{2} \) |
| 13 | \( 1 - 0.253T + 13T^{2} \) |
| 17 | \( 1 + 7.11T + 17T^{2} \) |
| 19 | \( 1 - 7.42T + 19T^{2} \) |
| 23 | \( 1 - 0.555T + 23T^{2} \) |
| 29 | \( 1 + 0.396T + 29T^{2} \) |
| 31 | \( 1 - 9.01T + 31T^{2} \) |
| 37 | \( 1 + 2.41T + 37T^{2} \) |
| 41 | \( 1 + 1.30T + 41T^{2} \) |
| 43 | \( 1 - 1.26T + 43T^{2} \) |
| 47 | \( 1 + 3.17T + 47T^{2} \) |
| 53 | \( 1 + 6.62T + 53T^{2} \) |
| 59 | \( 1 + 11.3T + 59T^{2} \) |
| 61 | \( 1 - 5.87T + 61T^{2} \) |
| 67 | \( 1 - 4.45T + 67T^{2} \) |
| 71 | \( 1 + 4.24T + 71T^{2} \) |
| 73 | \( 1 + 2.46T + 73T^{2} \) |
| 79 | \( 1 + 2.95T + 79T^{2} \) |
| 83 | \( 1 + 7.18T + 83T^{2} \) |
| 89 | \( 1 + 14.7T + 89T^{2} \) |
| 97 | \( 1 - 12.0T + 97T^{2} \) |
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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
−7.04996288262094122369803881596, −6.65775771221251475385632021787, −6.22077078016849272826286029351, −5.59398787825058603445725133801, −4.78072210707132302117694990034, −4.29872101311075621629790111807, −3.41799504757644445337190868039, −2.72794218539154599609059789452, −1.31476353321461730006657706945, 0,
1.31476353321461730006657706945, 2.72794218539154599609059789452, 3.41799504757644445337190868039, 4.29872101311075621629790111807, 4.78072210707132302117694990034, 5.59398787825058603445725133801, 6.22077078016849272826286029351, 6.65775771221251475385632021787, 7.04996288262094122369803881596