| L(s) = 1 | − 3-s + 5-s + 4.06·7-s + 9-s − 1.71·11-s + 13-s − 15-s + 6.06·17-s − 4.46·19-s − 4.06·21-s + 4.06·23-s + 25-s − 27-s − 2.46·29-s + 1.71·33-s + 4.06·35-s + 0.283·37-s − 39-s + 3.71·41-s + 6.81·43-s + 45-s + 9.52·49-s − 6.06·51-s + 7.09·53-s − 1.71·55-s + 4.46·57-s − 8.69·59-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 1.53·7-s + 0.333·9-s − 0.517·11-s + 0.277·13-s − 0.258·15-s + 1.47·17-s − 1.02·19-s − 0.887·21-s + 0.847·23-s + 0.200·25-s − 0.192·27-s − 0.457·29-s + 0.298·33-s + 0.687·35-s + 0.0465·37-s − 0.160·39-s + 0.580·41-s + 1.03·43-s + 0.149·45-s + 1.36·49-s − 0.849·51-s + 0.974·53-s − 0.231·55-s + 0.591·57-s − 1.13·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.344368656\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.344368656\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 - 4.06T + 7T^{2} \) |
| 11 | \( 1 + 1.71T + 11T^{2} \) |
| 17 | \( 1 - 6.06T + 17T^{2} \) |
| 19 | \( 1 + 4.46T + 19T^{2} \) |
| 23 | \( 1 - 4.06T + 23T^{2} \) |
| 29 | \( 1 + 2.46T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 0.283T + 37T^{2} \) |
| 41 | \( 1 - 3.71T + 41T^{2} \) |
| 43 | \( 1 - 6.81T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 7.09T + 53T^{2} \) |
| 59 | \( 1 + 8.69T + 59T^{2} \) |
| 61 | \( 1 + 7.84T + 61T^{2} \) |
| 67 | \( 1 - 6.24T + 67T^{2} \) |
| 71 | \( 1 + 3.03T + 71T^{2} \) |
| 73 | \( 1 - 8.34T + 73T^{2} \) |
| 79 | \( 1 - 3.60T + 79T^{2} \) |
| 83 | \( 1 - 10.2T + 83T^{2} \) |
| 89 | \( 1 - 5.60T + 89T^{2} \) |
| 97 | \( 1 + 2.86T + 97T^{2} \) |
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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
−7.82968039049446169136915618502, −7.57630937079469125465859442131, −6.52972429322234970133914014952, −5.76552238709150503569938119056, −5.22830926815233549107575266663, −4.64198698277079113267440294875, −3.76713429048629961683919500313, −2.60177561345319048228087210216, −1.70557675626931622003660204061, −0.880548242254232260419927789579,
0.880548242254232260419927789579, 1.70557675626931622003660204061, 2.60177561345319048228087210216, 3.76713429048629961683919500313, 4.64198698277079113267440294875, 5.22830926815233549107575266663, 5.76552238709150503569938119056, 6.52972429322234970133914014952, 7.57630937079469125465859442131, 7.82968039049446169136915618502
