| L(s) = 1 | − 2-s − 3-s + 4-s − 3.61·5-s + 6-s + 4.03·7-s − 8-s + 9-s + 3.61·10-s − 4.19·11-s − 12-s − 0.658·13-s − 4.03·14-s + 3.61·15-s + 16-s − 1.62·17-s − 18-s + 19-s − 3.61·20-s − 4.03·21-s + 4.19·22-s − 0.348·23-s + 24-s + 8.09·25-s + 0.658·26-s − 27-s + 4.03·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s − 1.61·5-s + 0.408·6-s + 1.52·7-s − 0.353·8-s + 0.333·9-s + 1.14·10-s − 1.26·11-s − 0.288·12-s − 0.182·13-s − 1.07·14-s + 0.934·15-s + 0.250·16-s − 0.394·17-s − 0.235·18-s + 0.229·19-s − 0.809·20-s − 0.880·21-s + 0.894·22-s − 0.0725·23-s + 0.204·24-s + 1.61·25-s + 0.129·26-s − 0.192·27-s + 0.762·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6042 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6042 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| 53 | \( 1 - T \) |
| good | 5 | \( 1 + 3.61T + 5T^{2} \) |
| 7 | \( 1 - 4.03T + 7T^{2} \) |
| 11 | \( 1 + 4.19T + 11T^{2} \) |
| 13 | \( 1 + 0.658T + 13T^{2} \) |
| 17 | \( 1 + 1.62T + 17T^{2} \) |
| 23 | \( 1 + 0.348T + 23T^{2} \) |
| 29 | \( 1 - 4.20T + 29T^{2} \) |
| 31 | \( 1 - 6.35T + 31T^{2} \) |
| 37 | \( 1 + 10.9T + 37T^{2} \) |
| 41 | \( 1 - 3.63T + 41T^{2} \) |
| 43 | \( 1 - 0.131T + 43T^{2} \) |
| 47 | \( 1 + 4.53T + 47T^{2} \) |
| 59 | \( 1 - 3.29T + 59T^{2} \) |
| 61 | \( 1 + 3.45T + 61T^{2} \) |
| 67 | \( 1 + 12.2T + 67T^{2} \) |
| 71 | \( 1 - 15.6T + 71T^{2} \) |
| 73 | \( 1 - 13.9T + 73T^{2} \) |
| 79 | \( 1 - 0.486T + 79T^{2} \) |
| 83 | \( 1 - 2.83T + 83T^{2} \) |
| 89 | \( 1 + 15.2T + 89T^{2} \) |
| 97 | \( 1 - 2.17T + 97T^{2} \) |
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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
−7.86810106186345950256522550271, −7.28036896751914915425173977894, −6.57329715233746356906399985139, −5.36988300580435183070068791914, −4.86135448692303682025813508880, −4.22670884564977560495778110209, −3.16725898874836422869221310501, −2.15595299699098995613376651395, −0.999121747361389695177722647443, 0,
0.999121747361389695177722647443, 2.15595299699098995613376651395, 3.16725898874836422869221310501, 4.22670884564977560495778110209, 4.86135448692303682025813508880, 5.36988300580435183070068791914, 6.57329715233746356906399985139, 7.28036896751914915425173977894, 7.86810106186345950256522550271
