| L(s) = 1 | − 5-s − 0.746·7-s − 3.14·11-s + 5.29·13-s + 3.89·17-s + 2.39·19-s − 4.74·23-s + 25-s + 8.43·29-s − 1.54·31-s + 0.746·35-s + 37-s + 5.94·41-s + 1.05·43-s − 7.69·47-s − 6.44·49-s − 9.52·53-s + 3.14·55-s − 14.3·59-s − 3.25·61-s − 5.29·65-s + 2.10·67-s − 1.89·71-s + 8.34·73-s + 2.34·77-s + 1.65·79-s − 4.48·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.282·7-s − 0.948·11-s + 1.46·13-s + 0.943·17-s + 0.550·19-s − 0.989·23-s + 0.200·25-s + 1.56·29-s − 0.277·31-s + 0.126·35-s + 0.164·37-s + 0.929·41-s + 0.161·43-s − 1.12·47-s − 0.920·49-s − 1.30·53-s + 0.424·55-s − 1.87·59-s − 0.416·61-s − 0.656·65-s + 0.257·67-s − 0.224·71-s + 0.977·73-s + 0.267·77-s + 0.185·79-s − 0.492·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6660 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6660 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.680970543\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.680970543\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 37 | \( 1 - T \) |
| good | 7 | \( 1 + 0.746T + 7T^{2} \) |
| 11 | \( 1 + 3.14T + 11T^{2} \) |
| 13 | \( 1 - 5.29T + 13T^{2} \) |
| 17 | \( 1 - 3.89T + 17T^{2} \) |
| 19 | \( 1 - 2.39T + 19T^{2} \) |
| 23 | \( 1 + 4.74T + 23T^{2} \) |
| 29 | \( 1 - 8.43T + 29T^{2} \) |
| 31 | \( 1 + 1.54T + 31T^{2} \) |
| 41 | \( 1 - 5.94T + 41T^{2} \) |
| 43 | \( 1 - 1.05T + 43T^{2} \) |
| 47 | \( 1 + 7.69T + 47T^{2} \) |
| 53 | \( 1 + 9.52T + 53T^{2} \) |
| 59 | \( 1 + 14.3T + 59T^{2} \) |
| 61 | \( 1 + 3.25T + 61T^{2} \) |
| 67 | \( 1 - 2.10T + 67T^{2} \) |
| 71 | \( 1 + 1.89T + 71T^{2} \) |
| 73 | \( 1 - 8.34T + 73T^{2} \) |
| 79 | \( 1 - 1.65T + 79T^{2} \) |
| 83 | \( 1 + 4.48T + 83T^{2} \) |
| 89 | \( 1 - 11.8T + 89T^{2} \) |
| 97 | \( 1 - 9.44T + 97T^{2} \) |
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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
−7.942069921377855316304278552101, −7.53515740356627235794947646107, −6.36973188187932161965252886915, −6.06143896137289474116362105979, −5.11031392600850411268994944973, −4.40355867018052614938083592059, −3.39888857640773071402418766012, −3.03381237319228867811077406120, −1.73588852762683329847278059561, −0.67907226344126537199095098779,
0.67907226344126537199095098779, 1.73588852762683329847278059561, 3.03381237319228867811077406120, 3.39888857640773071402418766012, 4.40355867018052614938083592059, 5.11031392600850411268994944973, 6.06143896137289474116362105979, 6.36973188187932161965252886915, 7.53515740356627235794947646107, 7.942069921377855316304278552101
