| L(s) = 1 | − 0.618·2-s − 3-s − 1.61·4-s − 5-s + 0.618·6-s + 3.23·7-s + 2.23·8-s + 9-s + 0.618·10-s + 1.61·12-s + 5.23·13-s − 2.00·14-s + 15-s + 1.85·16-s − 5.47·17-s − 0.618·18-s + 6.47·19-s + 1.61·20-s − 3.23·21-s − 4.70·23-s − 2.23·24-s + 25-s − 3.23·26-s − 27-s − 5.23·28-s − 1.23·29-s − 0.618·30-s + ⋯ |
| L(s) = 1 | − 0.437·2-s − 0.577·3-s − 0.809·4-s − 0.447·5-s + 0.252·6-s + 1.22·7-s + 0.790·8-s + 0.333·9-s + 0.195·10-s + 0.467·12-s + 1.45·13-s − 0.534·14-s + 0.258·15-s + 0.463·16-s − 1.32·17-s − 0.145·18-s + 1.48·19-s + 0.361·20-s − 0.706·21-s − 0.981·23-s − 0.456·24-s + 0.200·25-s − 0.634·26-s − 0.192·27-s − 0.989·28-s − 0.229·29-s − 0.112·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9736861355\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9736861355\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 0.618T + 2T^{2} \) |
| 7 | \( 1 - 3.23T + 7T^{2} \) |
| 13 | \( 1 - 5.23T + 13T^{2} \) |
| 17 | \( 1 + 5.47T + 17T^{2} \) |
| 19 | \( 1 - 6.47T + 19T^{2} \) |
| 23 | \( 1 + 4.70T + 23T^{2} \) |
| 29 | \( 1 + 1.23T + 29T^{2} \) |
| 31 | \( 1 + 6.70T + 31T^{2} \) |
| 37 | \( 1 - 0.763T + 37T^{2} \) |
| 41 | \( 1 - 3.52T + 41T^{2} \) |
| 43 | \( 1 + 5.23T + 43T^{2} \) |
| 47 | \( 1 - 8.70T + 47T^{2} \) |
| 53 | \( 1 - 9.94T + 53T^{2} \) |
| 59 | \( 1 - 11.7T + 59T^{2} \) |
| 61 | \( 1 + 1.47T + 61T^{2} \) |
| 67 | \( 1 - 11.2T + 67T^{2} \) |
| 71 | \( 1 + 14.4T + 71T^{2} \) |
| 73 | \( 1 + 10.4T + 73T^{2} \) |
| 79 | \( 1 - 12.7T + 79T^{2} \) |
| 83 | \( 1 + 4T + 83T^{2} \) |
| 89 | \( 1 - 4.76T + 89T^{2} \) |
| 97 | \( 1 - 12.7T + 97T^{2} \) |
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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
−9.037052855310290616352642500388, −8.582099751421481240773285062363, −7.79315117774579275858784669052, −7.13298187166834420041071977118, −5.87906373830787910810902376787, −5.19520931276398807688658271499, −4.30874286982828647610998319217, −3.70187628979622433491499798442, −1.83600078314410590847902660307, −0.78651131815844179878191431538,
0.78651131815844179878191431538, 1.83600078314410590847902660307, 3.70187628979622433491499798442, 4.30874286982828647610998319217, 5.19520931276398807688658271499, 5.87906373830787910810902376787, 7.13298187166834420041071977118, 7.79315117774579275858784669052, 8.582099751421481240773285062363, 9.037052855310290616352642500388
