| L(s) = 1 | + 2.73·2-s + 5.46·4-s − 5-s + 1.73·7-s + 9.46·8-s − 2.73·10-s + 2·11-s + 4.73·14-s + 14.9·16-s + 2.73·17-s − 7.46·19-s − 5.46·20-s + 5.46·22-s − 2·23-s + 25-s + 9.46·28-s + 6.73·29-s − 2.46·31-s + 21.8·32-s + 7.46·34-s − 1.73·35-s + 10.3·37-s − 20.3·38-s − 9.46·40-s + 7.26·41-s + 1.19·43-s + 10.9·44-s + ⋯ |
| L(s) = 1 | + 1.93·2-s + 2.73·4-s − 0.447·5-s + 0.654·7-s + 3.34·8-s − 0.863·10-s + 0.603·11-s + 1.26·14-s + 3.73·16-s + 0.662·17-s − 1.71·19-s − 1.22·20-s + 1.16·22-s − 0.417·23-s + 0.200·25-s + 1.78·28-s + 1.25·29-s − 0.442·31-s + 3.86·32-s + 1.28·34-s − 0.292·35-s + 1.70·37-s − 3.30·38-s − 1.49·40-s + 1.13·41-s + 0.182·43-s + 1.64·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(8.512944398\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.512944398\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 2.73T + 2T^{2} \) |
| 7 | \( 1 - 1.73T + 7T^{2} \) |
| 11 | \( 1 - 2T + 11T^{2} \) |
| 17 | \( 1 - 2.73T + 17T^{2} \) |
| 19 | \( 1 + 7.46T + 19T^{2} \) |
| 23 | \( 1 + 2T + 23T^{2} \) |
| 29 | \( 1 - 6.73T + 29T^{2} \) |
| 31 | \( 1 + 2.46T + 31T^{2} \) |
| 37 | \( 1 - 10.3T + 37T^{2} \) |
| 41 | \( 1 - 7.26T + 41T^{2} \) |
| 43 | \( 1 - 1.19T + 43T^{2} \) |
| 47 | \( 1 - 10.1T + 47T^{2} \) |
| 53 | \( 1 - 2.53T + 53T^{2} \) |
| 59 | \( 1 + 5.66T + 59T^{2} \) |
| 61 | \( 1 + 15.3T + 61T^{2} \) |
| 67 | \( 1 - 1.19T + 67T^{2} \) |
| 71 | \( 1 - 1.26T + 71T^{2} \) |
| 73 | \( 1 + 1.73T + 73T^{2} \) |
| 79 | \( 1 + 11T + 79T^{2} \) |
| 83 | \( 1 - 10.9T + 83T^{2} \) |
| 89 | \( 1 + 8.73T + 89T^{2} \) |
| 97 | \( 1 + 5.19T + 97T^{2} \) |
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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
−7.66953391992331562424364770645, −6.96589015274545583989533853131, −6.11604238703808822805906094063, −5.89066181758862648938485211970, −4.72758886036947140190259886712, −4.41049416741754075180361649232, −3.82191203361696358535235572879, −2.89122947557770006024249452439, −2.17394129426177312157774826844, −1.18492094197526191441500866806,
1.18492094197526191441500866806, 2.17394129426177312157774826844, 2.89122947557770006024249452439, 3.82191203361696358535235572879, 4.41049416741754075180361649232, 4.72758886036947140190259886712, 5.89066181758862648938485211970, 6.11604238703808822805906094063, 6.96589015274545583989533853131, 7.66953391992331562424364770645
