L(s) = 1 | − 1.09·2-s − 0.223·3-s − 0.800·4-s + 0.245·6-s − 2.26·7-s + 3.06·8-s − 2.94·9-s − 3.16·11-s + 0.179·12-s + 0.0649·13-s + 2.47·14-s − 1.75·16-s − 4.44·17-s + 3.23·18-s + 7.82·19-s + 0.506·21-s + 3.46·22-s + 2.53·23-s − 0.686·24-s − 0.0711·26-s + 1.33·27-s + 1.81·28-s − 6.53·29-s + 8.27·31-s − 4.20·32-s + 0.709·33-s + 4.86·34-s + ⋯ |
L(s) = 1 | − 0.774·2-s − 0.129·3-s − 0.400·4-s + 0.100·6-s − 0.855·7-s + 1.08·8-s − 0.983·9-s − 0.954·11-s + 0.0517·12-s + 0.0180·13-s + 0.662·14-s − 0.439·16-s − 1.07·17-s + 0.761·18-s + 1.79·19-s + 0.110·21-s + 0.739·22-s + 0.527·23-s − 0.140·24-s − 0.0139·26-s + 0.256·27-s + 0.342·28-s − 1.21·29-s + 1.48·31-s − 0.744·32-s + 0.123·33-s + 0.834·34-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 | \( 1 + 1.09T + 2T^{2} \) |
| 3 | \( 1 + 0.223T + 3T^{2} \) |
| 7 | \( 1 + 2.26T + 7T^{2} \) |
| 11 | \( 1 + 3.16T + 11T^{2} \) |
| 13 | \( 1 - 0.0649T + 13T^{2} \) |
| 17 | \( 1 + 4.44T + 17T^{2} \) |
| 19 | \( 1 - 7.82T + 19T^{2} \) |
| 23 | \( 1 - 2.53T + 23T^{2} \) |
| 29 | \( 1 + 6.53T + 29T^{2} \) |
| 31 | \( 1 - 8.27T + 31T^{2} \) |
| 37 | \( 1 - 2.73T + 37T^{2} \) |
| 41 | \( 1 + 7.52T + 41T^{2} \) |
| 43 | \( 1 - 10.7T + 43T^{2} \) |
| 47 | \( 1 + 8.53T + 47T^{2} \) |
| 53 | \( 1 + 0.0654T + 53T^{2} \) |
| 59 | \( 1 - 11.9T + 59T^{2} \) |
| 61 | \( 1 - 7.88T + 61T^{2} \) |
| 67 | \( 1 - 13.3T + 67T^{2} \) |
| 71 | \( 1 + 8.56T + 71T^{2} \) |
| 73 | \( 1 - 16.6T + 73T^{2} \) |
| 79 | \( 1 + 0.725T + 79T^{2} \) |
| 83 | \( 1 - 6.57T + 83T^{2} \) |
| 89 | \( 1 + 10.8T + 89T^{2} \) |
| 97 | \( 1 - 9.64T + 97T^{2} \) |
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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
−7.969399745707588987742254563605, −7.11141948018389680980895998727, −6.45736490346984409662596834750, −5.38355271356636235257423216862, −5.11587821034922205750368166174, −3.97434678570362253451772658224, −3.10640564100693777130520441893, −2.34390942653616083823419031935, −0.908318805671807676462270208885, 0,
0.908318805671807676462270208885, 2.34390942653616083823419031935, 3.10640564100693777130520441893, 3.97434678570362253451772658224, 5.11587821034922205750368166174, 5.38355271356636235257423216862, 6.45736490346984409662596834750, 7.11141948018389680980895998727, 7.969399745707588987742254563605