| L(s) = 1 | + 0.767·2-s + 0.193·3-s − 1.41·4-s − 0.953·5-s + 0.148·6-s − 2.43·7-s − 2.61·8-s − 2.96·9-s − 0.732·10-s − 1.86·11-s − 0.273·12-s − 0.753·13-s − 1.87·14-s − 0.184·15-s + 0.809·16-s − 2.32·17-s − 2.27·18-s + 8.15·19-s + 1.34·20-s − 0.471·21-s − 1.43·22-s + 2.32·23-s − 0.507·24-s − 4.09·25-s − 0.578·26-s − 1.15·27-s + 3.43·28-s + ⋯ |
| L(s) = 1 | + 0.543·2-s + 0.111·3-s − 0.705·4-s − 0.426·5-s + 0.0607·6-s − 0.920·7-s − 0.925·8-s − 0.987·9-s − 0.231·10-s − 0.563·11-s − 0.0788·12-s − 0.209·13-s − 0.500·14-s − 0.0476·15-s + 0.202·16-s − 0.563·17-s − 0.536·18-s + 1.86·19-s + 0.300·20-s − 0.102·21-s − 0.306·22-s + 0.484·23-s − 0.103·24-s − 0.818·25-s − 0.113·26-s − 0.222·27-s + 0.649·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 349 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 349 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 349 | \( 1 + T \) |
| good | 2 | \( 1 - 0.767T + 2T^{2} \) |
| 3 | \( 1 - 0.193T + 3T^{2} \) |
| 5 | \( 1 + 0.953T + 5T^{2} \) |
| 7 | \( 1 + 2.43T + 7T^{2} \) |
| 11 | \( 1 + 1.86T + 11T^{2} \) |
| 13 | \( 1 + 0.753T + 13T^{2} \) |
| 17 | \( 1 + 2.32T + 17T^{2} \) |
| 19 | \( 1 - 8.15T + 19T^{2} \) |
| 23 | \( 1 - 2.32T + 23T^{2} \) |
| 29 | \( 1 + 1.04T + 29T^{2} \) |
| 31 | \( 1 + 9.69T + 31T^{2} \) |
| 37 | \( 1 - 6.87T + 37T^{2} \) |
| 41 | \( 1 + 4.63T + 41T^{2} \) |
| 43 | \( 1 + 4.27T + 43T^{2} \) |
| 47 | \( 1 + 9.35T + 47T^{2} \) |
| 53 | \( 1 - 13.0T + 53T^{2} \) |
| 59 | \( 1 - 5.63T + 59T^{2} \) |
| 61 | \( 1 - 1.18T + 61T^{2} \) |
| 67 | \( 1 + 8.45T + 67T^{2} \) |
| 71 | \( 1 + 7.10T + 71T^{2} \) |
| 73 | \( 1 - 4.70T + 73T^{2} \) |
| 79 | \( 1 - 5.13T + 79T^{2} \) |
| 83 | \( 1 + 15.3T + 83T^{2} \) |
| 89 | \( 1 + 7.74T + 89T^{2} \) |
| 97 | \( 1 + 1.00T + 97T^{2} \) |
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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
−11.30871346697893831444924868326, −9.899781104342375358617072026527, −9.243586854502722003029832576732, −8.283384202933944737512621335603, −7.20536862894307245615602228632, −5.84903304800204235842758193550, −5.12388050923333605252557523467, −3.71295428605922164991452401969, −2.92557239626741201570375630471, 0,
2.92557239626741201570375630471, 3.71295428605922164991452401969, 5.12388050923333605252557523467, 5.84903304800204235842758193550, 7.20536862894307245615602228632, 8.283384202933944737512621335603, 9.243586854502722003029832576732, 9.899781104342375358617072026527, 11.30871346697893831444924868326
