| L(s) = 1 | − 0.652·3-s + 3.53·5-s − 2.57·9-s − 11-s + 4.41·13-s − 2.30·15-s − 5.24·17-s − 1.81·19-s + 6.33·23-s + 7.47·25-s + 3.63·27-s + 1.92·29-s − 1.46·31-s + 0.652·33-s + 4.45·37-s − 2.87·39-s + 0.283·41-s + 3.41·43-s − 9.09·45-s + 4.55·47-s + 3.42·51-s − 7.23·53-s − 3.53·55-s + 1.18·57-s − 9.53·59-s − 1.14·61-s + 15.5·65-s + ⋯ |
| L(s) = 1 | − 0.376·3-s + 1.57·5-s − 0.857·9-s − 0.301·11-s + 1.22·13-s − 0.595·15-s − 1.27·17-s − 0.416·19-s + 1.32·23-s + 1.49·25-s + 0.700·27-s + 0.356·29-s − 0.263·31-s + 0.113·33-s + 0.732·37-s − 0.461·39-s + 0.0442·41-s + 0.520·43-s − 1.35·45-s + 0.664·47-s + 0.479·51-s − 0.993·53-s − 0.476·55-s + 0.156·57-s − 1.24·59-s − 0.146·61-s + 1.93·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8624 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.326515668\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.326515668\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 3 | \( 1 + 0.652T + 3T^{2} \) |
| 5 | \( 1 - 3.53T + 5T^{2} \) |
| 13 | \( 1 - 4.41T + 13T^{2} \) |
| 17 | \( 1 + 5.24T + 17T^{2} \) |
| 19 | \( 1 + 1.81T + 19T^{2} \) |
| 23 | \( 1 - 6.33T + 23T^{2} \) |
| 29 | \( 1 - 1.92T + 29T^{2} \) |
| 31 | \( 1 + 1.46T + 31T^{2} \) |
| 37 | \( 1 - 4.45T + 37T^{2} \) |
| 41 | \( 1 - 0.283T + 41T^{2} \) |
| 43 | \( 1 - 3.41T + 43T^{2} \) |
| 47 | \( 1 - 4.55T + 47T^{2} \) |
| 53 | \( 1 + 7.23T + 53T^{2} \) |
| 59 | \( 1 + 9.53T + 59T^{2} \) |
| 61 | \( 1 + 1.14T + 61T^{2} \) |
| 67 | \( 1 - 0.694T + 67T^{2} \) |
| 71 | \( 1 + 9.46T + 71T^{2} \) |
| 73 | \( 1 - 2.34T + 73T^{2} \) |
| 79 | \( 1 - 12.4T + 79T^{2} \) |
| 83 | \( 1 - 11.3T + 83T^{2} \) |
| 89 | \( 1 + 3.46T + 89T^{2} \) |
| 97 | \( 1 - 15.3T + 97T^{2} \) |
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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
−7.80016241941916893516068166260, −6.77047681580405847260881010290, −6.22552931583518655965086558785, −5.90016033981521143150761275330, −5.10460757431989594780853382601, −4.48983444580934834316905375266, −3.28269459202678104769072532392, −2.55859391835418662471798371483, −1.80345748737411265601389880453, −0.76225224717283288100997784823,
0.76225224717283288100997784823, 1.80345748737411265601389880453, 2.55859391835418662471798371483, 3.28269459202678104769072532392, 4.48983444580934834316905375266, 5.10460757431989594780853382601, 5.90016033981521143150761275330, 6.22552931583518655965086558785, 6.77047681580405847260881010290, 7.80016241941916893516068166260
