| L(s) = 1 | − 0.561·5-s − 4.56·7-s − 11-s − 6.12·13-s + 5.56·17-s + 0.561·19-s − 4.68·23-s − 4.68·25-s + 6.56·29-s − 0.438·31-s + 2.56·35-s + 1.12·37-s − 0.684·41-s + 43-s + 6.56·47-s + 13.8·49-s + 6.43·53-s + 0.561·55-s − 4·59-s − 6.24·61-s + 3.43·65-s + 0.438·67-s − 3.12·71-s + 8·73-s + 4.56·77-s + 2.87·79-s − 5·83-s + ⋯ |
| L(s) = 1 | − 0.251·5-s − 1.72·7-s − 0.301·11-s − 1.69·13-s + 1.34·17-s + 0.128·19-s − 0.976·23-s − 0.936·25-s + 1.21·29-s − 0.0787·31-s + 0.432·35-s + 0.184·37-s − 0.106·41-s + 0.152·43-s + 0.957·47-s + 1.97·49-s + 0.884·53-s + 0.0757·55-s − 0.520·59-s − 0.799·61-s + 0.426·65-s + 0.0535·67-s − 0.370·71-s + 0.936·73-s + 0.519·77-s + 0.323·79-s − 0.548·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3096 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3096 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8723085322\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8723085322\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 43 | \( 1 - T \) |
| good | 5 | \( 1 + 0.561T + 5T^{2} \) |
| 7 | \( 1 + 4.56T + 7T^{2} \) |
| 11 | \( 1 + T + 11T^{2} \) |
| 13 | \( 1 + 6.12T + 13T^{2} \) |
| 17 | \( 1 - 5.56T + 17T^{2} \) |
| 19 | \( 1 - 0.561T + 19T^{2} \) |
| 23 | \( 1 + 4.68T + 23T^{2} \) |
| 29 | \( 1 - 6.56T + 29T^{2} \) |
| 31 | \( 1 + 0.438T + 31T^{2} \) |
| 37 | \( 1 - 1.12T + 37T^{2} \) |
| 41 | \( 1 + 0.684T + 41T^{2} \) |
| 47 | \( 1 - 6.56T + 47T^{2} \) |
| 53 | \( 1 - 6.43T + 53T^{2} \) |
| 59 | \( 1 + 4T + 59T^{2} \) |
| 61 | \( 1 + 6.24T + 61T^{2} \) |
| 67 | \( 1 - 0.438T + 67T^{2} \) |
| 71 | \( 1 + 3.12T + 71T^{2} \) |
| 73 | \( 1 - 8T + 73T^{2} \) |
| 79 | \( 1 - 2.87T + 79T^{2} \) |
| 83 | \( 1 + 5T + 83T^{2} \) |
| 89 | \( 1 - 8.24T + 89T^{2} \) |
| 97 | \( 1 - 18.3T + 97T^{2} \) |
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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
−8.780058034973417486734310454207, −7.67301627531586097644608529546, −7.39101026132541475010360310646, −6.37443720969574530215501302356, −5.78202222219561010558044274326, −4.85723271538012431994975118536, −3.83942029020910973647968251897, −3.07826875395510413354863186106, −2.30022675951325734831949830595, −0.53263568651362052684770248510,
0.53263568651362052684770248510, 2.30022675951325734831949830595, 3.07826875395510413354863186106, 3.83942029020910973647968251897, 4.85723271538012431994975118536, 5.78202222219561010558044274326, 6.37443720969574530215501302356, 7.39101026132541475010360310646, 7.67301627531586097644608529546, 8.780058034973417486734310454207
