L(s) = 1 | + 0.720·2-s − 1.48·4-s + 3.36·5-s + 0.451·7-s − 2.50·8-s + 2.42·10-s − 5.21·11-s − 5.16·13-s + 0.325·14-s + 1.15·16-s − 7.12·17-s − 0.740·19-s − 4.97·20-s − 3.75·22-s − 7.26·23-s + 6.30·25-s − 3.71·26-s − 0.668·28-s − 0.897·29-s + 6.36·31-s + 5.84·32-s − 5.13·34-s + 1.51·35-s + 0.200·37-s − 0.533·38-s − 8.42·40-s + 5.33·41-s + ⋯ |
L(s) = 1 | + 0.509·2-s − 0.740·4-s + 1.50·5-s + 0.170·7-s − 0.886·8-s + 0.765·10-s − 1.57·11-s − 1.43·13-s + 0.0869·14-s + 0.289·16-s − 1.72·17-s − 0.169·19-s − 1.11·20-s − 0.801·22-s − 1.51·23-s + 1.26·25-s − 0.729·26-s − 0.126·28-s − 0.166·29-s + 1.14·31-s + 1.03·32-s − 0.880·34-s + 0.256·35-s + 0.0329·37-s − 0.0865·38-s − 1.33·40-s + 0.832·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1341 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1341 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 149 | \( 1 - T \) |
good | 2 | \( 1 - 0.720T + 2T^{2} \) |
| 5 | \( 1 - 3.36T + 5T^{2} \) |
| 7 | \( 1 - 0.451T + 7T^{2} \) |
| 11 | \( 1 + 5.21T + 11T^{2} \) |
| 13 | \( 1 + 5.16T + 13T^{2} \) |
| 17 | \( 1 + 7.12T + 17T^{2} \) |
| 19 | \( 1 + 0.740T + 19T^{2} \) |
| 23 | \( 1 + 7.26T + 23T^{2} \) |
| 29 | \( 1 + 0.897T + 29T^{2} \) |
| 31 | \( 1 - 6.36T + 31T^{2} \) |
| 37 | \( 1 - 0.200T + 37T^{2} \) |
| 41 | \( 1 - 5.33T + 41T^{2} \) |
| 43 | \( 1 - 6.16T + 43T^{2} \) |
| 47 | \( 1 + 8.69T + 47T^{2} \) |
| 53 | \( 1 - 12.3T + 53T^{2} \) |
| 59 | \( 1 + 6.95T + 59T^{2} \) |
| 61 | \( 1 + 2.55T + 61T^{2} \) |
| 67 | \( 1 - 2.74T + 67T^{2} \) |
| 71 | \( 1 + 8.88T + 71T^{2} \) |
| 73 | \( 1 - 3.10T + 73T^{2} \) |
| 79 | \( 1 + 0.613T + 79T^{2} \) |
| 83 | \( 1 + 0.114T + 83T^{2} \) |
| 89 | \( 1 + 7.68T + 89T^{2} \) |
| 97 | \( 1 + 6.73T + 97T^{2} \) |
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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
−9.367509409696202539750700933947, −8.500109718307398974416765202537, −7.63904839074778235925342075187, −6.42574606910644023461879571387, −5.72833483057947295609683320480, −4.95582937805243594151900890561, −4.38221863299898151259524952396, −2.71268641267436813445848240771, −2.16468302587930389000777193169, 0,
2.16468302587930389000777193169, 2.71268641267436813445848240771, 4.38221863299898151259524952396, 4.95582937805243594151900890561, 5.72833483057947295609683320480, 6.42574606910644023461879571387, 7.63904839074778235925342075187, 8.500109718307398974416765202537, 9.367509409696202539750700933947