| L(s) = 1 | − 3-s − 5-s + 2.95·7-s + 9-s + 4·11-s + 4.95·13-s + 15-s + 0.421·17-s − 5.06·19-s − 2.95·21-s + 7.48·23-s + 25-s − 27-s + 8.44·29-s − 31-s − 4·33-s − 2.95·35-s + 4.11·37-s − 4.95·39-s − 12.1·41-s + 1.06·43-s − 45-s + 6.64·47-s + 1.73·49-s − 0.421·51-s − 3.57·53-s − 4·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 1.11·7-s + 0.333·9-s + 1.20·11-s + 1.37·13-s + 0.258·15-s + 0.102·17-s − 1.16·19-s − 0.644·21-s + 1.56·23-s + 0.200·25-s − 0.192·27-s + 1.56·29-s − 0.179·31-s − 0.696·33-s − 0.499·35-s + 0.676·37-s − 0.793·39-s − 1.89·41-s + 0.162·43-s − 0.149·45-s + 0.969·47-s + 0.247·49-s − 0.0590·51-s − 0.491·53-s − 0.539·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.007706570\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.007706570\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 31 | \( 1 + T \) |
| good | 7 | \( 1 - 2.95T + 7T^{2} \) |
| 11 | \( 1 - 4T + 11T^{2} \) |
| 13 | \( 1 - 4.95T + 13T^{2} \) |
| 17 | \( 1 - 0.421T + 17T^{2} \) |
| 19 | \( 1 + 5.06T + 19T^{2} \) |
| 23 | \( 1 - 7.48T + 23T^{2} \) |
| 29 | \( 1 - 8.44T + 29T^{2} \) |
| 37 | \( 1 - 4.11T + 37T^{2} \) |
| 41 | \( 1 + 12.1T + 41T^{2} \) |
| 43 | \( 1 - 1.06T + 43T^{2} \) |
| 47 | \( 1 - 6.64T + 47T^{2} \) |
| 53 | \( 1 + 3.57T + 53T^{2} \) |
| 59 | \( 1 + 0.534T + 59T^{2} \) |
| 61 | \( 1 - 3.91T + 61T^{2} \) |
| 67 | \( 1 + 8.02T + 67T^{2} \) |
| 71 | \( 1 - 3.46T + 71T^{2} \) |
| 73 | \( 1 + 8.11T + 73T^{2} \) |
| 79 | \( 1 + 3.48T + 79T^{2} \) |
| 83 | \( 1 + 1.80T + 83T^{2} \) |
| 89 | \( 1 - 7.60T + 89T^{2} \) |
| 97 | \( 1 + 18.0T + 97T^{2} \) |
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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
−8.641778366689497969771998352573, −7.86257143401722579990587659516, −6.82738714243230785922084062195, −6.45303306214654406598695459435, −5.49941719735364674580747788190, −4.60014588150219937990208269205, −4.12820344070545622014612126270, −3.11842989425214552146938726617, −1.65943997627586665328910056825, −0.948987686877017089214317486225,
0.948987686877017089214317486225, 1.65943997627586665328910056825, 3.11842989425214552146938726617, 4.12820344070545622014612126270, 4.60014588150219937990208269205, 5.49941719735364674580747788190, 6.45303306214654406598695459435, 6.82738714243230785922084062195, 7.86257143401722579990587659516, 8.641778366689497969771998352573
