| L(s) = 1 | + 3-s − 2.49·5-s + 4.04·7-s + 9-s + 4.82·11-s + 4.29·13-s − 2.49·15-s − 4.51·17-s + 4.43·19-s + 4.04·21-s + 1.20·25-s + 27-s − 0.499·29-s + 5.11·31-s + 4.82·33-s − 10.0·35-s + 10.6·37-s + 4.29·39-s − 7.60·41-s + 7.58·43-s − 2.49·45-s − 7.67·47-s + 9.36·49-s − 4.51·51-s + 7.63·53-s − 12.0·55-s + 4.43·57-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.11·5-s + 1.52·7-s + 0.333·9-s + 1.45·11-s + 1.19·13-s − 0.643·15-s − 1.09·17-s + 1.01·19-s + 0.882·21-s + 0.240·25-s + 0.192·27-s − 0.0926·29-s + 0.918·31-s + 0.840·33-s − 1.70·35-s + 1.75·37-s + 0.687·39-s − 1.18·41-s + 1.15·43-s − 0.371·45-s − 1.11·47-s + 1.33·49-s − 0.631·51-s + 1.04·53-s − 1.62·55-s + 0.586·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6348 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6348 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.084899139\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.084899139\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 23 | \( 1 \) |
| good | 5 | \( 1 + 2.49T + 5T^{2} \) |
| 7 | \( 1 - 4.04T + 7T^{2} \) |
| 11 | \( 1 - 4.82T + 11T^{2} \) |
| 13 | \( 1 - 4.29T + 13T^{2} \) |
| 17 | \( 1 + 4.51T + 17T^{2} \) |
| 19 | \( 1 - 4.43T + 19T^{2} \) |
| 29 | \( 1 + 0.499T + 29T^{2} \) |
| 31 | \( 1 - 5.11T + 31T^{2} \) |
| 37 | \( 1 - 10.6T + 37T^{2} \) |
| 41 | \( 1 + 7.60T + 41T^{2} \) |
| 43 | \( 1 - 7.58T + 43T^{2} \) |
| 47 | \( 1 + 7.67T + 47T^{2} \) |
| 53 | \( 1 - 7.63T + 53T^{2} \) |
| 59 | \( 1 - 2.82T + 59T^{2} \) |
| 61 | \( 1 + 1.58T + 61T^{2} \) |
| 67 | \( 1 + 12.0T + 67T^{2} \) |
| 71 | \( 1 - 3.79T + 71T^{2} \) |
| 73 | \( 1 + 7.45T + 73T^{2} \) |
| 79 | \( 1 - 11.6T + 79T^{2} \) |
| 83 | \( 1 + 5.30T + 83T^{2} \) |
| 89 | \( 1 + 11.1T + 89T^{2} \) |
| 97 | \( 1 + 7.13T + 97T^{2} \) |
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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
−8.081875741353701113869411293142, −7.52698905797367622639039797810, −6.77022002073795419499318051610, −6.00086904101248865702847094568, −4.90372976698059379311536569266, −4.16825320229590122311921526734, −3.90064813641032247886785021903, −2.84501128128630949100873335062, −1.66166249865008688746147765810, −0.996047744145417379682111311490,
0.996047744145417379682111311490, 1.66166249865008688746147765810, 2.84501128128630949100873335062, 3.90064813641032247886785021903, 4.16825320229590122311921526734, 4.90372976698059379311536569266, 6.00086904101248865702847094568, 6.77022002073795419499318051610, 7.52698905797367622639039797810, 8.081875741353701113869411293142
