L(s) = 1 | − 2.51·3-s − 0.472·5-s − 4.16·7-s + 3.33·9-s + 0.304·11-s + 3.93·13-s + 1.18·15-s + 5.23·17-s + 3.05·19-s + 10.4·21-s − 7.82·23-s − 4.77·25-s − 0.847·27-s − 7.79·29-s + 9.12·31-s − 0.766·33-s + 1.96·35-s − 7.45·37-s − 9.90·39-s − 5.32·41-s − 10.2·43-s − 1.57·45-s − 8.20·47-s + 10.3·49-s − 13.1·51-s − 8.29·53-s − 0.143·55-s + ⋯ |
L(s) = 1 | − 1.45·3-s − 0.211·5-s − 1.57·7-s + 1.11·9-s + 0.0918·11-s + 1.09·13-s + 0.306·15-s + 1.26·17-s + 0.701·19-s + 2.28·21-s − 1.63·23-s − 0.955·25-s − 0.163·27-s − 1.44·29-s + 1.63·31-s − 0.133·33-s + 0.332·35-s − 1.22·37-s − 1.58·39-s − 0.832·41-s − 1.56·43-s − 0.234·45-s − 1.19·47-s + 1.48·49-s − 1.84·51-s − 1.13·53-s − 0.0193·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5210939888\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5210939888\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 251 | \( 1 - T \) |
good | 3 | \( 1 + 2.51T + 3T^{2} \) |
| 5 | \( 1 + 0.472T + 5T^{2} \) |
| 7 | \( 1 + 4.16T + 7T^{2} \) |
| 11 | \( 1 - 0.304T + 11T^{2} \) |
| 13 | \( 1 - 3.93T + 13T^{2} \) |
| 17 | \( 1 - 5.23T + 17T^{2} \) |
| 19 | \( 1 - 3.05T + 19T^{2} \) |
| 23 | \( 1 + 7.82T + 23T^{2} \) |
| 29 | \( 1 + 7.79T + 29T^{2} \) |
| 31 | \( 1 - 9.12T + 31T^{2} \) |
| 37 | \( 1 + 7.45T + 37T^{2} \) |
| 41 | \( 1 + 5.32T + 41T^{2} \) |
| 43 | \( 1 + 10.2T + 43T^{2} \) |
| 47 | \( 1 + 8.20T + 47T^{2} \) |
| 53 | \( 1 + 8.29T + 53T^{2} \) |
| 59 | \( 1 + 3.03T + 59T^{2} \) |
| 61 | \( 1 + 0.845T + 61T^{2} \) |
| 67 | \( 1 - 5.94T + 67T^{2} \) |
| 71 | \( 1 + 6.20T + 71T^{2} \) |
| 73 | \( 1 - 10.1T + 73T^{2} \) |
| 79 | \( 1 - 9.74T + 79T^{2} \) |
| 83 | \( 1 - 4.98T + 83T^{2} \) |
| 89 | \( 1 - 12.3T + 89T^{2} \) |
| 97 | \( 1 - 0.396T + 97T^{2} \) |
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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
−8.322995174592809097836845231488, −7.62679887765525807362543099805, −6.58002135914844531897585252729, −6.26356665687670636220234712520, −5.67336510896210008075171735313, −4.89737398604161338799445599463, −3.59889303564163328176107010930, −3.41755500332368348417355932655, −1.67390349094417220839153283864, −0.44690094835506484034641559343,
0.44690094835506484034641559343, 1.67390349094417220839153283864, 3.41755500332368348417355932655, 3.59889303564163328176107010930, 4.89737398604161338799445599463, 5.67336510896210008075171735313, 6.26356665687670636220234712520, 6.58002135914844531897585252729, 7.62679887765525807362543099805, 8.322995174592809097836845231488