| L(s) = 1 | + 2.21·3-s − 3.58·5-s − 5.12·7-s + 1.92·9-s + 3.17·11-s − 13-s − 7.95·15-s − 0.575·17-s − 0.413·19-s − 11.3·21-s + 1.60·23-s + 7.86·25-s − 2.39·27-s + 29-s − 5.28·31-s + 7.03·33-s + 18.3·35-s − 4.88·37-s − 2.21·39-s − 8.50·41-s − 9.21·43-s − 6.88·45-s + 0.918·47-s + 19.2·49-s − 1.27·51-s + 7.87·53-s − 11.3·55-s + ⋯ |
| L(s) = 1 | + 1.28·3-s − 1.60·5-s − 1.93·7-s + 0.640·9-s + 0.955·11-s − 0.277·13-s − 2.05·15-s − 0.139·17-s − 0.0947·19-s − 2.48·21-s + 0.334·23-s + 1.57·25-s − 0.460·27-s + 0.185·29-s − 0.948·31-s + 1.22·33-s + 3.10·35-s − 0.803·37-s − 0.355·39-s − 1.32·41-s − 1.40·43-s − 1.02·45-s + 0.134·47-s + 2.75·49-s − 0.178·51-s + 1.08·53-s − 1.53·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.181769832\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.181769832\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 13 | \( 1 + T \) |
| 29 | \( 1 - T \) |
| good | 3 | \( 1 - 2.21T + 3T^{2} \) |
| 5 | \( 1 + 3.58T + 5T^{2} \) |
| 7 | \( 1 + 5.12T + 7T^{2} \) |
| 11 | \( 1 - 3.17T + 11T^{2} \) |
| 17 | \( 1 + 0.575T + 17T^{2} \) |
| 19 | \( 1 + 0.413T + 19T^{2} \) |
| 23 | \( 1 - 1.60T + 23T^{2} \) |
| 31 | \( 1 + 5.28T + 31T^{2} \) |
| 37 | \( 1 + 4.88T + 37T^{2} \) |
| 41 | \( 1 + 8.50T + 41T^{2} \) |
| 43 | \( 1 + 9.21T + 43T^{2} \) |
| 47 | \( 1 - 0.918T + 47T^{2} \) |
| 53 | \( 1 - 7.87T + 53T^{2} \) |
| 59 | \( 1 + 8.70T + 59T^{2} \) |
| 61 | \( 1 - 5.62T + 61T^{2} \) |
| 67 | \( 1 - 13.1T + 67T^{2} \) |
| 71 | \( 1 - 9.39T + 71T^{2} \) |
| 73 | \( 1 + 0.559T + 73T^{2} \) |
| 79 | \( 1 - 1.65T + 79T^{2} \) |
| 83 | \( 1 - 10.4T + 83T^{2} \) |
| 89 | \( 1 - 7.12T + 89T^{2} \) |
| 97 | \( 1 - 10.0T + 97T^{2} \) |
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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
−8.232664377562851695464301266647, −7.26114549912175254184553582515, −6.96927432057567387767828320207, −6.23961872048926122499388866110, −5.01211566662508873295330576724, −3.91983690542856424863743114493, −3.52113708606203421466538246444, −3.19687535048294877228915079087, −2.11081401146039261827803354658, −0.50690402575002748553135849601,
0.50690402575002748553135849601, 2.11081401146039261827803354658, 3.19687535048294877228915079087, 3.52113708606203421466538246444, 3.91983690542856424863743114493, 5.01211566662508873295330576724, 6.23961872048926122499388866110, 6.96927432057567387767828320207, 7.26114549912175254184553582515, 8.232664377562851695464301266647
