| L(s) = 1 | + 2.44·3-s + 5-s + 2·7-s + 2.99·9-s + 0.449·11-s − 13-s + 2.44·15-s − 2.89·17-s − 4.44·19-s + 4.89·21-s + 1.55·23-s + 25-s + 4·29-s + 0.449·31-s + 1.10·33-s + 2·35-s − 4.89·37-s − 2.44·39-s + 1.10·41-s − 3.34·43-s + 2.99·45-s − 2·47-s − 3·49-s − 7.10·51-s + 10.8·53-s + 0.449·55-s − 10.8·57-s + ⋯ |
| L(s) = 1 | + 1.41·3-s + 0.447·5-s + 0.755·7-s + 0.999·9-s + 0.135·11-s − 0.277·13-s + 0.632·15-s − 0.703·17-s − 1.02·19-s + 1.06·21-s + 0.323·23-s + 0.200·25-s + 0.742·29-s + 0.0807·31-s + 0.191·33-s + 0.338·35-s − 0.805·37-s − 0.392·39-s + 0.171·41-s − 0.510·43-s + 0.447·45-s − 0.291·47-s − 0.428·49-s − 0.994·51-s + 1.49·53-s + 0.0606·55-s − 1.44·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.473673598\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.473673598\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| good | 3 | \( 1 - 2.44T + 3T^{2} \) |
| 7 | \( 1 - 2T + 7T^{2} \) |
| 11 | \( 1 - 0.449T + 11T^{2} \) |
| 17 | \( 1 + 2.89T + 17T^{2} \) |
| 19 | \( 1 + 4.44T + 19T^{2} \) |
| 23 | \( 1 - 1.55T + 23T^{2} \) |
| 29 | \( 1 - 4T + 29T^{2} \) |
| 31 | \( 1 - 0.449T + 31T^{2} \) |
| 37 | \( 1 + 4.89T + 37T^{2} \) |
| 41 | \( 1 - 1.10T + 41T^{2} \) |
| 43 | \( 1 + 3.34T + 43T^{2} \) |
| 47 | \( 1 + 2T + 47T^{2} \) |
| 53 | \( 1 - 10.8T + 53T^{2} \) |
| 59 | \( 1 + 5.34T + 59T^{2} \) |
| 61 | \( 1 - 13.7T + 61T^{2} \) |
| 67 | \( 1 + 14.8T + 67T^{2} \) |
| 71 | \( 1 + 8.44T + 71T^{2} \) |
| 73 | \( 1 - 14.6T + 73T^{2} \) |
| 79 | \( 1 + 4.89T + 79T^{2} \) |
| 83 | \( 1 - 2T + 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 + 11.7T + 97T^{2} \) |
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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
−10.70525758672027089757299013829, −9.855490631365725808039136213883, −8.828004100374299851941654892946, −8.468772761594242766495781029739, −7.45021289498554912174980419510, −6.45042225654760342330317057616, −5.04198320919738406846616321643, −4.01831226697534092421927954877, −2.72510253910502481469015184697, −1.80293423917785422604569313169,
1.80293423917785422604569313169, 2.72510253910502481469015184697, 4.01831226697534092421927954877, 5.04198320919738406846616321643, 6.45042225654760342330317057616, 7.45021289498554912174980419510, 8.468772761594242766495781029739, 8.828004100374299851941654892946, 9.855490631365725808039136213883, 10.70525758672027089757299013829
