| L(s) = 1 | − 1.30·2-s − 0.302·4-s + 4.30·5-s − 7-s + 3·8-s − 5.60·10-s − 1.30·11-s + 13-s + 1.30·14-s − 3.30·16-s + 4.30·17-s − 5.30·19-s − 1.30·20-s + 1.69·22-s + 3.90·23-s + 13.5·25-s − 1.30·26-s + 0.302·28-s + 0.394·29-s + 7.21·31-s − 1.69·32-s − 5.60·34-s − 4.30·35-s + 4.21·37-s + 6.90·38-s + 12.9·40-s + 9·41-s + ⋯ |
| L(s) = 1 | − 0.921·2-s − 0.151·4-s + 1.92·5-s − 0.377·7-s + 1.06·8-s − 1.77·10-s − 0.392·11-s + 0.277·13-s + 0.348·14-s − 0.825·16-s + 1.04·17-s − 1.21·19-s − 0.291·20-s + 0.361·22-s + 0.814·23-s + 2.70·25-s − 0.255·26-s + 0.0572·28-s + 0.0732·29-s + 1.29·31-s − 0.300·32-s − 0.961·34-s − 0.727·35-s + 0.692·37-s + 1.12·38-s + 2.04·40-s + 1.40·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 351 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 351 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.027852864\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.027852864\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 2 | \( 1 + 1.30T + 2T^{2} \) |
| 5 | \( 1 - 4.30T + 5T^{2} \) |
| 7 | \( 1 + T + 7T^{2} \) |
| 11 | \( 1 + 1.30T + 11T^{2} \) |
| 17 | \( 1 - 4.30T + 17T^{2} \) |
| 19 | \( 1 + 5.30T + 19T^{2} \) |
| 23 | \( 1 - 3.90T + 23T^{2} \) |
| 29 | \( 1 - 0.394T + 29T^{2} \) |
| 31 | \( 1 - 7.21T + 31T^{2} \) |
| 37 | \( 1 - 4.21T + 37T^{2} \) |
| 41 | \( 1 - 9T + 41T^{2} \) |
| 43 | \( 1 - 0.697T + 43T^{2} \) |
| 47 | \( 1 + 0.908T + 47T^{2} \) |
| 53 | \( 1 + 12.9T + 53T^{2} \) |
| 59 | \( 1 - 3T + 59T^{2} \) |
| 61 | \( 1 - 2.39T + 61T^{2} \) |
| 67 | \( 1 + 13.5T + 67T^{2} \) |
| 71 | \( 1 - 13.8T + 71T^{2} \) |
| 73 | \( 1 + 10.9T + 73T^{2} \) |
| 79 | \( 1 + 13.9T + 79T^{2} \) |
| 83 | \( 1 + 0.394T + 83T^{2} \) |
| 89 | \( 1 - 8.60T + 89T^{2} \) |
| 97 | \( 1 + 14.4T + 97T^{2} \) |
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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
−10.96632760512361187435883141703, −10.23686429166615985935362348652, −9.657874541498972227081295703374, −8.958084264753605810503780161478, −7.983282360637784219511460652492, −6.64389701921348176929445455401, −5.78398706958049095128073790077, −4.67324741992451976949119887348, −2.70744784226701525052801596168, −1.31726451116100674971788723911,
1.31726451116100674971788723911, 2.70744784226701525052801596168, 4.67324741992451976949119887348, 5.78398706958049095128073790077, 6.64389701921348176929445455401, 7.983282360637784219511460652492, 8.958084264753605810503780161478, 9.657874541498972227081295703374, 10.23686429166615985935362348652, 10.96632760512361187435883141703
