L(s) = 1 | − 1.61·2-s + 3-s + 0.618·4-s + 5-s − 1.61·6-s − 7-s + 2.23·8-s + 9-s − 1.61·10-s + 0.618·12-s + 5.47·13-s + 1.61·14-s + 15-s − 4.85·16-s − 0.763·17-s − 1.61·18-s − 6.70·19-s + 0.618·20-s − 21-s − 7.70·23-s + 2.23·24-s − 4·25-s − 8.85·26-s + 27-s − 0.618·28-s − 5·29-s − 1.61·30-s + ⋯ |
L(s) = 1 | − 1.14·2-s + 0.577·3-s + 0.309·4-s + 0.447·5-s − 0.660·6-s − 0.377·7-s + 0.790·8-s + 0.333·9-s − 0.511·10-s + 0.178·12-s + 1.51·13-s + 0.432·14-s + 0.258·15-s − 1.21·16-s − 0.185·17-s − 0.381·18-s − 1.53·19-s + 0.138·20-s − 0.218·21-s − 1.60·23-s + 0.456·24-s − 0.800·25-s − 1.73·26-s + 0.192·27-s − 0.116·28-s − 0.928·29-s − 0.295·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2541 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2541 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + 1.61T + 2T^{2} \) |
| 5 | \( 1 - T + 5T^{2} \) |
| 13 | \( 1 - 5.47T + 13T^{2} \) |
| 17 | \( 1 + 0.763T + 17T^{2} \) |
| 19 | \( 1 + 6.70T + 19T^{2} \) |
| 23 | \( 1 + 7.70T + 23T^{2} \) |
| 29 | \( 1 + 5T + 29T^{2} \) |
| 31 | \( 1 + 0.763T + 31T^{2} \) |
| 37 | \( 1 + 7T + 37T^{2} \) |
| 41 | \( 1 + 6.47T + 41T^{2} \) |
| 43 | \( 1 - 7.70T + 43T^{2} \) |
| 47 | \( 1 + 4.23T + 47T^{2} \) |
| 53 | \( 1 - 10.1T + 53T^{2} \) |
| 59 | \( 1 - 11.1T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 + 14.2T + 67T^{2} \) |
| 71 | \( 1 - 6.47T + 71T^{2} \) |
| 73 | \( 1 + 13.4T + 73T^{2} \) |
| 79 | \( 1 - 5.52T + 79T^{2} \) |
| 83 | \( 1 + 11.2T + 83T^{2} \) |
| 89 | \( 1 - 4.47T + 89T^{2} \) |
| 97 | \( 1 + 3.70T + 97T^{2} \) |
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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
−8.631661670961521898740278317530, −8.102509404520733915056448017343, −7.20600364294757369658462743057, −6.36082936808633132103586205250, −5.64620076840498920306806671098, −4.22141299721480089949822197099, −3.71466294802770733815741198019, −2.22562801411153565200756508523, −1.56027752106596086333411163762, 0,
1.56027752106596086333411163762, 2.22562801411153565200756508523, 3.71466294802770733815741198019, 4.22141299721480089949822197099, 5.64620076840498920306806671098, 6.36082936808633132103586205250, 7.20600364294757369658462743057, 8.102509404520733915056448017343, 8.631661670961521898740278317530