| L(s) = 1 | + 2.15·3-s − 3.75·5-s + 1.62·9-s + 4.89·11-s + 1.80·13-s − 8.07·15-s + 4.27·17-s + 5.31·19-s + 0.795·23-s + 9.10·25-s − 2.95·27-s + 0.466·29-s − 2.71·31-s + 10.5·33-s − 10.1·37-s + 3.87·39-s − 41-s − 3.98·43-s − 6.11·45-s + 2.29·47-s + 9.19·51-s − 1.15·53-s − 18.3·55-s + 11.4·57-s + 6.87·59-s + 8.97·61-s − 6.76·65-s + ⋯ |
| L(s) = 1 | + 1.24·3-s − 1.67·5-s + 0.542·9-s + 1.47·11-s + 0.499·13-s − 2.08·15-s + 1.03·17-s + 1.21·19-s + 0.165·23-s + 1.82·25-s − 0.568·27-s + 0.0865·29-s − 0.487·31-s + 1.83·33-s − 1.66·37-s + 0.620·39-s − 0.156·41-s − 0.608·43-s − 0.911·45-s + 0.335·47-s + 1.28·51-s − 0.158·53-s − 2.47·55-s + 1.51·57-s + 0.894·59-s + 1.14·61-s − 0.839·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8036 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8036 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.722658692\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.722658692\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 41 | \( 1 + T \) |
| good | 3 | \( 1 - 2.15T + 3T^{2} \) |
| 5 | \( 1 + 3.75T + 5T^{2} \) |
| 11 | \( 1 - 4.89T + 11T^{2} \) |
| 13 | \( 1 - 1.80T + 13T^{2} \) |
| 17 | \( 1 - 4.27T + 17T^{2} \) |
| 19 | \( 1 - 5.31T + 19T^{2} \) |
| 23 | \( 1 - 0.795T + 23T^{2} \) |
| 29 | \( 1 - 0.466T + 29T^{2} \) |
| 31 | \( 1 + 2.71T + 31T^{2} \) |
| 37 | \( 1 + 10.1T + 37T^{2} \) |
| 43 | \( 1 + 3.98T + 43T^{2} \) |
| 47 | \( 1 - 2.29T + 47T^{2} \) |
| 53 | \( 1 + 1.15T + 53T^{2} \) |
| 59 | \( 1 - 6.87T + 59T^{2} \) |
| 61 | \( 1 - 8.97T + 61T^{2} \) |
| 67 | \( 1 + 3.33T + 67T^{2} \) |
| 71 | \( 1 - 0.810T + 71T^{2} \) |
| 73 | \( 1 + 2.12T + 73T^{2} \) |
| 79 | \( 1 + 6.00T + 79T^{2} \) |
| 83 | \( 1 + 4.93T + 83T^{2} \) |
| 89 | \( 1 - 13.0T + 89T^{2} \) |
| 97 | \( 1 - 12.0T + 97T^{2} \) |
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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
−7.83373213810021571377927310047, −7.33730242074233230023117010296, −6.81390863909949922526729298578, −5.73088013920283929815370170095, −4.80966072024346413699447768053, −3.83765597855077756300588903075, −3.55672187130205757052512091773, −3.09063368762476464123836590911, −1.74662331036333516192599014268, −0.803020480788443876137356714188,
0.803020480788443876137356714188, 1.74662331036333516192599014268, 3.09063368762476464123836590911, 3.55672187130205757052512091773, 3.83765597855077756300588903075, 4.80966072024346413699447768053, 5.73088013920283929815370170095, 6.81390863909949922526729298578, 7.33730242074233230023117010296, 7.83373213810021571377927310047
