| L(s) = 1 | − 1.73·2-s + 0.999·4-s + 5-s + 1.73·8-s − 3·9-s − 1.73·10-s + 1.46·11-s + 2·13-s − 5·16-s − 3.46·17-s + 5.19·18-s − 2·19-s + 0.999·20-s − 2.53·22-s − 6.92·23-s + 25-s − 3.46·26-s + 29-s − 4.92·31-s + 5.19·32-s + 5.99·34-s − 2.99·36-s + 3.46·38-s + 1.73·40-s + 1.46·41-s − 3.46·43-s + 1.46·44-s + ⋯ |
| L(s) = 1 | − 1.22·2-s + 0.499·4-s + 0.447·5-s + 0.612·8-s − 9-s − 0.547·10-s + 0.441·11-s + 0.554·13-s − 1.25·16-s − 0.840·17-s + 1.22·18-s − 0.458·19-s + 0.223·20-s − 0.540·22-s − 1.44·23-s + 0.200·25-s − 0.679·26-s + 0.185·29-s − 0.885·31-s + 0.918·32-s + 1.02·34-s − 0.499·36-s + 0.561·38-s + 0.273·40-s + 0.228·41-s − 0.528·43-s + 0.220·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7105 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6691363758\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6691363758\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| 29 | \( 1 - T \) |
| good | 2 | \( 1 + 1.73T + 2T^{2} \) |
| 3 | \( 1 + 3T^{2} \) |
| 11 | \( 1 - 1.46T + 11T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 + 3.46T + 17T^{2} \) |
| 19 | \( 1 + 2T + 19T^{2} \) |
| 23 | \( 1 + 6.92T + 23T^{2} \) |
| 31 | \( 1 + 4.92T + 31T^{2} \) |
| 37 | \( 1 + 37T^{2} \) |
| 41 | \( 1 - 1.46T + 41T^{2} \) |
| 43 | \( 1 + 3.46T + 43T^{2} \) |
| 47 | \( 1 - 4T + 47T^{2} \) |
| 53 | \( 1 + 8.92T + 53T^{2} \) |
| 59 | \( 1 - 8T + 59T^{2} \) |
| 61 | \( 1 + 5.46T + 61T^{2} \) |
| 67 | \( 1 + 4T + 67T^{2} \) |
| 71 | \( 1 + 8T + 71T^{2} \) |
| 73 | \( 1 - 6.39T + 73T^{2} \) |
| 79 | \( 1 - 5.46T + 79T^{2} \) |
| 83 | \( 1 - 14.9T + 83T^{2} \) |
| 89 | \( 1 - 10.5T + 89T^{2} \) |
| 97 | \( 1 - 4.53T + 97T^{2} \) |
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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
−8.091893689260667357912453835623, −7.50866116450495643441454210584, −6.47717323364258879772992099514, −6.16001574631107038626418748651, −5.19114364622544366948955309729, −4.32943011806704704915471897479, −3.49752688860009174779579485625, −2.31064101302835281633468553459, −1.71991611475297263492075794104, −0.49615108747465581667819550554,
0.49615108747465581667819550554, 1.71991611475297263492075794104, 2.31064101302835281633468553459, 3.49752688860009174779579485625, 4.32943011806704704915471897479, 5.19114364622544366948955309729, 6.16001574631107038626418748651, 6.47717323364258879772992099514, 7.50866116450495643441454210584, 8.091893689260667357912453835623
