| L(s) = 1 | + 1.56·3-s + 3.12·7-s − 0.561·9-s − 4·11-s − 3.56·13-s − 5.12·17-s + 4·19-s + 4.87·21-s − 23-s − 5.56·27-s − 4.43·29-s + 5.56·31-s − 6.24·33-s − 1.12·37-s − 5.56·39-s − 3.56·41-s + 0.876·43-s − 8.68·47-s + 2.75·49-s − 8·51-s − 12.2·53-s + 6.24·57-s + 10.2·59-s + 2.87·61-s − 1.75·63-s + 10.2·67-s − 1.56·69-s + ⋯ |
| L(s) = 1 | + 0.901·3-s + 1.18·7-s − 0.187·9-s − 1.20·11-s − 0.987·13-s − 1.24·17-s + 0.917·19-s + 1.06·21-s − 0.208·23-s − 1.07·27-s − 0.824·29-s + 0.998·31-s − 1.08·33-s − 0.184·37-s − 0.890·39-s − 0.556·41-s + 0.133·43-s − 1.26·47-s + 0.393·49-s − 1.12·51-s − 1.68·53-s + 0.827·57-s + 1.33·59-s + 0.368·61-s − 0.220·63-s + 1.25·67-s − 0.187·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 3 | \( 1 - 1.56T + 3T^{2} \) |
| 7 | \( 1 - 3.12T + 7T^{2} \) |
| 11 | \( 1 + 4T + 11T^{2} \) |
| 13 | \( 1 + 3.56T + 13T^{2} \) |
| 17 | \( 1 + 5.12T + 17T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 29 | \( 1 + 4.43T + 29T^{2} \) |
| 31 | \( 1 - 5.56T + 31T^{2} \) |
| 37 | \( 1 + 1.12T + 37T^{2} \) |
| 41 | \( 1 + 3.56T + 41T^{2} \) |
| 43 | \( 1 - 0.876T + 43T^{2} \) |
| 47 | \( 1 + 8.68T + 47T^{2} \) |
| 53 | \( 1 + 12.2T + 53T^{2} \) |
| 59 | \( 1 - 10.2T + 59T^{2} \) |
| 61 | \( 1 - 2.87T + 61T^{2} \) |
| 67 | \( 1 - 10.2T + 67T^{2} \) |
| 71 | \( 1 + 8.68T + 71T^{2} \) |
| 73 | \( 1 + 12.4T + 73T^{2} \) |
| 79 | \( 1 - 6.24T + 79T^{2} \) |
| 83 | \( 1 + 12T + 83T^{2} \) |
| 89 | \( 1 - 10T + 89T^{2} \) |
| 97 | \( 1 + 0.246T + 97T^{2} \) |
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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
−8.043263029068261767989856582225, −7.52566892838366889086445582171, −6.68985401511206780447490330070, −5.49260115746555568794419120038, −5.01160260568204496222600984846, −4.25250360972244496520702328841, −3.10897367761189709719639502997, −2.45736458521699049868109411435, −1.71510230203107351462532255140, 0,
1.71510230203107351462532255140, 2.45736458521699049868109411435, 3.10897367761189709719639502997, 4.25250360972244496520702328841, 5.01160260568204496222600984846, 5.49260115746555568794419120038, 6.68985401511206780447490330070, 7.52566892838366889086445582171, 8.043263029068261767989856582225
