L(s) = 1 | − 1.61·2-s − 3-s + 0.618·4-s + 1.61·6-s − 7-s + 2.23·8-s + 9-s + 11-s − 0.618·12-s + 5.47·13-s + 1.61·14-s − 4.85·16-s − 0.763·17-s − 1.61·18-s + 6.70·19-s + 21-s − 1.61·22-s + 7.70·23-s − 2.23·24-s − 8.85·26-s − 27-s − 0.618·28-s + 5·29-s − 0.763·31-s + 3.38·32-s − 33-s + 1.23·34-s + ⋯ |
L(s) = 1 | − 1.14·2-s − 0.577·3-s + 0.309·4-s + 0.660·6-s − 0.377·7-s + 0.790·8-s + 0.333·9-s + 0.301·11-s − 0.178·12-s + 1.51·13-s + 0.432·14-s − 1.21·16-s − 0.185·17-s − 0.381·18-s + 1.53·19-s + 0.218·21-s − 0.344·22-s + 1.60·23-s − 0.456·24-s − 1.73·26-s − 0.192·27-s − 0.116·28-s + 0.928·29-s − 0.137·31-s + 0.597·32-s − 0.174·33-s + 0.211·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5775 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5775 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.065992230\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.065992230\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 - T \) |
good | 2 | \( 1 + 1.61T + 2T^{2} \) |
| 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
−8.208520660486485694787371629714, −7.45552283948838077710272353991, −6.85663049370700925035941373323, −6.12206141581698848591972037019, −5.35422831630737534132414259856, −4.48211294095250253063893064027, −3.67682321554640306035754501984, −2.65668727198678965554497583395, −1.18871771788384588926628286648, −0.861402166253848872451509356303,
0.861402166253848872451509356303, 1.18871771788384588926628286648, 2.65668727198678965554497583395, 3.67682321554640306035754501984, 4.48211294095250253063893064027, 5.35422831630737534132414259856, 6.12206141581698848591972037019, 6.85663049370700925035941373323, 7.45552283948838077710272353991, 8.208520660486485694787371629714