L(s) = 1 | − 2.35·2-s + 1.07·3-s + 3.52·4-s + 3.29·5-s − 2.52·6-s + 3.84·7-s − 3.59·8-s − 1.84·9-s − 7.75·10-s − 3.17·11-s + 3.79·12-s + 13-s − 9.05·14-s + 3.54·15-s + 1.39·16-s + 0.572·17-s + 4.34·18-s − 6.09·19-s + 11.6·20-s + 4.13·21-s + 7.46·22-s + 8.94·23-s − 3.86·24-s + 5.87·25-s − 2.35·26-s − 5.20·27-s + 13.5·28-s + ⋯ |
L(s) = 1 | − 1.66·2-s + 0.620·3-s + 1.76·4-s + 1.47·5-s − 1.03·6-s + 1.45·7-s − 1.27·8-s − 0.615·9-s − 2.45·10-s − 0.957·11-s + 1.09·12-s + 0.277·13-s − 2.41·14-s + 0.914·15-s + 0.348·16-s + 0.138·17-s + 1.02·18-s − 1.39·19-s + 2.60·20-s + 0.902·21-s + 1.59·22-s + 1.86·23-s − 0.788·24-s + 1.17·25-s − 0.461·26-s − 1.00·27-s + 2.56·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1339 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1339 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.351650937\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.351650937\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 13 | \( 1 - T \) |
| 103 | \( 1 + T \) |
good | 2 | \( 1 + 2.35T + 2T^{2} \) |
| 3 | \( 1 - 1.07T + 3T^{2} \) |
| 5 | \( 1 - 3.29T + 5T^{2} \) |
| 7 | \( 1 - 3.84T + 7T^{2} \) |
| 11 | \( 1 + 3.17T + 11T^{2} \) |
| 17 | \( 1 - 0.572T + 17T^{2} \) |
| 19 | \( 1 + 6.09T + 19T^{2} \) |
| 23 | \( 1 - 8.94T + 23T^{2} \) |
| 29 | \( 1 - 7.43T + 29T^{2} \) |
| 31 | \( 1 + 2.51T + 31T^{2} \) |
| 37 | \( 1 - 6.12T + 37T^{2} \) |
| 41 | \( 1 - 6.84T + 41T^{2} \) |
| 43 | \( 1 + 3.57T + 43T^{2} \) |
| 47 | \( 1 - 8.72T + 47T^{2} \) |
| 53 | \( 1 - 6.83T + 53T^{2} \) |
| 59 | \( 1 + 9.41T + 59T^{2} \) |
| 61 | \( 1 + 10.9T + 61T^{2} \) |
| 67 | \( 1 - 13.1T + 67T^{2} \) |
| 71 | \( 1 - 1.60T + 71T^{2} \) |
| 73 | \( 1 - 7.15T + 73T^{2} \) |
| 79 | \( 1 + 4.65T + 79T^{2} \) |
| 83 | \( 1 - 2.45T + 83T^{2} \) |
| 89 | \( 1 - 0.362T + 89T^{2} \) |
| 97 | \( 1 + 2.74T + 97T^{2} \) |
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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
−9.322782502508407299325461210944, −8.862648305824626746817822205914, −8.241320823394223001911391720735, −7.63331393231378715379137824481, −6.55698512010057340246263322081, −5.62949094721821203265361168032, −4.70767768593096509024727392910, −2.67475381792004393680435647518, −2.19591348363137141299267099607, −1.12752477597494398207527936655,
1.12752477597494398207527936655, 2.19591348363137141299267099607, 2.67475381792004393680435647518, 4.70767768593096509024727392910, 5.62949094721821203265361168032, 6.55698512010057340246263322081, 7.63331393231378715379137824481, 8.241320823394223001911391720735, 8.862648305824626746817822205914, 9.322782502508407299325461210944