| L(s) = 1 | + 3-s + 1.63·5-s + 0.968·7-s + 9-s − 0.515·11-s + 2.41·13-s + 1.63·15-s + 2.72·17-s + 3.28·19-s + 0.968·21-s + 2.42·23-s − 2.33·25-s + 27-s + 2.20·29-s − 6.45·31-s − 0.515·33-s + 1.58·35-s + 4.93·37-s + 2.41·39-s − 1.19·41-s + 11.4·43-s + 1.63·45-s + 0.331·47-s − 6.06·49-s + 2.72·51-s + 0.641·53-s − 0.841·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.729·5-s + 0.366·7-s + 0.333·9-s − 0.155·11-s + 0.670·13-s + 0.421·15-s + 0.661·17-s + 0.754·19-s + 0.211·21-s + 0.505·23-s − 0.467·25-s + 0.192·27-s + 0.409·29-s − 1.15·31-s − 0.0898·33-s + 0.267·35-s + 0.811·37-s + 0.386·39-s − 0.187·41-s + 1.74·43-s + 0.243·45-s + 0.0483·47-s − 0.865·49-s + 0.381·51-s + 0.0880·53-s − 0.113·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6036 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6036 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.431312201\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.431312201\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 503 | \( 1 - T \) |
| good | 5 | \( 1 - 1.63T + 5T^{2} \) |
| 7 | \( 1 - 0.968T + 7T^{2} \) |
| 11 | \( 1 + 0.515T + 11T^{2} \) |
| 13 | \( 1 - 2.41T + 13T^{2} \) |
| 17 | \( 1 - 2.72T + 17T^{2} \) |
| 19 | \( 1 - 3.28T + 19T^{2} \) |
| 23 | \( 1 - 2.42T + 23T^{2} \) |
| 29 | \( 1 - 2.20T + 29T^{2} \) |
| 31 | \( 1 + 6.45T + 31T^{2} \) |
| 37 | \( 1 - 4.93T + 37T^{2} \) |
| 41 | \( 1 + 1.19T + 41T^{2} \) |
| 43 | \( 1 - 11.4T + 43T^{2} \) |
| 47 | \( 1 - 0.331T + 47T^{2} \) |
| 53 | \( 1 - 0.641T + 53T^{2} \) |
| 59 | \( 1 + 2.73T + 59T^{2} \) |
| 61 | \( 1 - 3.89T + 61T^{2} \) |
| 67 | \( 1 - 5.15T + 67T^{2} \) |
| 71 | \( 1 + 1.21T + 71T^{2} \) |
| 73 | \( 1 - 3.96T + 73T^{2} \) |
| 79 | \( 1 + 15.3T + 79T^{2} \) |
| 83 | \( 1 - 8.12T + 83T^{2} \) |
| 89 | \( 1 + 2.00T + 89T^{2} \) |
| 97 | \( 1 + 3.76T + 97T^{2} \) |
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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
−8.002347300964992467014076473020, −7.53795256544251564570410385821, −6.69368241010520030426722007041, −5.82071555257602717771840763903, −5.35321818173356686100695079240, −4.38526162442967163790725326768, −3.54904623570646753433370399742, −2.76529761773648197032459084923, −1.85920693624399906230685302757, −1.01409725614559813060241479791,
1.01409725614559813060241479791, 1.85920693624399906230685302757, 2.76529761773648197032459084923, 3.54904623570646753433370399742, 4.38526162442967163790725326768, 5.35321818173356686100695079240, 5.82071555257602717771840763903, 6.69368241010520030426722007041, 7.53795256544251564570410385821, 8.002347300964992467014076473020
