| L(s) = 1 | + 1.90·2-s − 6.07·3-s − 4.36·4-s − 8.17·5-s − 11.5·6-s − 17.8·7-s − 23.5·8-s + 9.95·9-s − 15.5·10-s + 11·11-s + 26.5·12-s − 34.1·14-s + 49.7·15-s − 9.96·16-s − 63.0·17-s + 18.9·18-s + 98.2·19-s + 35.7·20-s + 108.·21-s + 20.9·22-s − 1.03·23-s + 143.·24-s − 58.0·25-s + 103.·27-s + 78.1·28-s − 44.6·29-s + 94.7·30-s + ⋯ |
| L(s) = 1 | + 0.673·2-s − 1.16·3-s − 0.546·4-s − 0.731·5-s − 0.788·6-s − 0.966·7-s − 1.04·8-s + 0.368·9-s − 0.492·10-s + 0.301·11-s + 0.638·12-s − 0.651·14-s + 0.855·15-s − 0.155·16-s − 0.899·17-s + 0.248·18-s + 1.18·19-s + 0.399·20-s + 1.13·21-s + 0.203·22-s − 0.00933·23-s + 1.21·24-s − 0.464·25-s + 0.738·27-s + 0.527·28-s − 0.285·29-s + 0.576·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 - 11T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 1.90T + 8T^{2} \) |
| 3 | \( 1 + 6.07T + 27T^{2} \) |
| 5 | \( 1 + 8.17T + 125T^{2} \) |
| 7 | \( 1 + 17.8T + 343T^{2} \) |
| 17 | \( 1 + 63.0T + 4.91e3T^{2} \) |
| 19 | \( 1 - 98.2T + 6.85e3T^{2} \) |
| 23 | \( 1 + 1.03T + 1.21e4T^{2} \) |
| 29 | \( 1 + 44.6T + 2.43e4T^{2} \) |
| 31 | \( 1 + 112.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 241.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 404.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 20.2T + 7.95e4T^{2} \) |
| 47 | \( 1 - 49.8T + 1.03e5T^{2} \) |
| 53 | \( 1 - 158.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 503.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 10.6T + 2.26e5T^{2} \) |
| 67 | \( 1 - 584.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 382.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 855.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 239.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.17e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.30e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.31e3T + 9.12e5T^{2} \) |
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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
−8.556659803906979521527575614962, −7.48718506825415861474149154554, −6.62776999216945040151538027548, −5.90621425973237524311325840235, −5.31921783102972142490269503583, −4.34048440090447148769953350222, −3.74314205959822434350565717376, −2.74670609831799497064087455496, −0.78823470175013337473649095842, 0,
0.78823470175013337473649095842, 2.74670609831799497064087455496, 3.74314205959822434350565717376, 4.34048440090447148769953350222, 5.31921783102972142490269503583, 5.90621425973237524311325840235, 6.62776999216945040151538027548, 7.48718506825415861474149154554, 8.556659803906979521527575614962
