| L(s) = 1 | + 3-s − 5-s − 0.508·7-s + 9-s + 5.36·11-s − 13-s − 15-s + 7.87·17-s − 0.508·19-s − 0.508·21-s + 1.87·23-s + 25-s + 27-s + 3.87·29-s − 6.37·31-s + 5.36·33-s + 0.508·35-s − 9.74·37-s − 39-s − 0.983·41-s + 8·43-s − 45-s + 4.34·47-s − 6.74·49-s + 7.87·51-s + 3.01·53-s − 5.36·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s − 0.192·7-s + 0.333·9-s + 1.61·11-s − 0.277·13-s − 0.258·15-s + 1.90·17-s − 0.116·19-s − 0.110·21-s + 0.390·23-s + 0.200·25-s + 0.192·27-s + 0.718·29-s − 1.14·31-s + 0.933·33-s + 0.0859·35-s − 1.60·37-s − 0.160·39-s − 0.153·41-s + 1.21·43-s − 0.149·45-s + 0.633·47-s − 0.963·49-s + 1.10·51-s + 0.414·53-s − 0.723·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.707467482\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.707467482\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 + 0.508T + 7T^{2} \) |
| 11 | \( 1 - 5.36T + 11T^{2} \) |
| 17 | \( 1 - 7.87T + 17T^{2} \) |
| 19 | \( 1 + 0.508T + 19T^{2} \) |
| 23 | \( 1 - 1.87T + 23T^{2} \) |
| 29 | \( 1 - 3.87T + 29T^{2} \) |
| 31 | \( 1 + 6.37T + 31T^{2} \) |
| 37 | \( 1 + 9.74T + 37T^{2} \) |
| 41 | \( 1 + 0.983T + 41T^{2} \) |
| 43 | \( 1 - 8T + 43T^{2} \) |
| 47 | \( 1 - 4.34T + 47T^{2} \) |
| 53 | \( 1 - 3.01T + 53T^{2} \) |
| 59 | \( 1 - 13.3T + 59T^{2} \) |
| 61 | \( 1 - 6T + 61T^{2} \) |
| 67 | \( 1 + 2.37T + 67T^{2} \) |
| 71 | \( 1 + 13.1T + 71T^{2} \) |
| 73 | \( 1 + 7.87T + 73T^{2} \) |
| 79 | \( 1 - 12.7T + 79T^{2} \) |
| 83 | \( 1 + 17.1T + 83T^{2} \) |
| 89 | \( 1 - 2T + 89T^{2} \) |
| 97 | \( 1 - 15.6T + 97T^{2} \) |
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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
−8.051302593412627626035322717617, −7.27850911512937588343112707218, −6.88764941696653451347841636084, −5.92962810630583086735812269001, −5.18459123862801400854555601399, −4.16779830899275782097279009509, −3.61437793620380128878053056968, −2.98340823659141697814418580726, −1.75962336567073341323044963813, −0.886965499427696958083909288269,
0.886965499427696958083909288269, 1.75962336567073341323044963813, 2.98340823659141697814418580726, 3.61437793620380128878053056968, 4.16779830899275782097279009509, 5.18459123862801400854555601399, 5.92962810630583086735812269001, 6.88764941696653451347841636084, 7.27850911512937588343112707218, 8.051302593412627626035322717617
