| L(s) = 1 | + 0.414·2-s − 1.82·4-s − 5-s − 1.58·8-s − 0.414·10-s − 4.82·11-s + 0.828·13-s + 3·16-s + 0.828·17-s − 2.82·19-s + 1.82·20-s − 1.99·22-s + 2.41·23-s + 25-s + 0.343·26-s + 29-s − 6·31-s + 4.41·32-s + 0.343·34-s − 1.17·38-s + 1.58·40-s + 2.17·41-s + 6.41·43-s + 8.82·44-s + 0.999·46-s − 2·47-s + 0.414·50-s + ⋯ |
| L(s) = 1 | + 0.292·2-s − 0.914·4-s − 0.447·5-s − 0.560·8-s − 0.130·10-s − 1.45·11-s + 0.229·13-s + 0.750·16-s + 0.200·17-s − 0.648·19-s + 0.408·20-s − 0.426·22-s + 0.503·23-s + 0.200·25-s + 0.0672·26-s + 0.185·29-s − 1.07·31-s + 0.780·32-s + 0.0588·34-s − 0.190·38-s + 0.250·40-s + 0.339·41-s + 0.978·43-s + 1.33·44-s + 0.147·46-s − 0.291·47-s + 0.0585·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.038680700\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.038680700\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - 0.414T + 2T^{2} \) |
| 11 | \( 1 + 4.82T + 11T^{2} \) |
| 13 | \( 1 - 0.828T + 13T^{2} \) |
| 17 | \( 1 - 0.828T + 17T^{2} \) |
| 19 | \( 1 + 2.82T + 19T^{2} \) |
| 23 | \( 1 - 2.41T + 23T^{2} \) |
| 29 | \( 1 - T + 29T^{2} \) |
| 31 | \( 1 + 6T + 31T^{2} \) |
| 37 | \( 1 + 37T^{2} \) |
| 41 | \( 1 - 2.17T + 41T^{2} \) |
| 43 | \( 1 - 6.41T + 43T^{2} \) |
| 47 | \( 1 + 2T + 47T^{2} \) |
| 53 | \( 1 - 6.82T + 53T^{2} \) |
| 59 | \( 1 - 12.4T + 59T^{2} \) |
| 61 | \( 1 + 11.4T + 61T^{2} \) |
| 67 | \( 1 - 12.4T + 67T^{2} \) |
| 71 | \( 1 - 12.4T + 71T^{2} \) |
| 73 | \( 1 - 4.82T + 73T^{2} \) |
| 79 | \( 1 - 9.17T + 79T^{2} \) |
| 83 | \( 1 - 11.7T + 83T^{2} \) |
| 89 | \( 1 + 2.65T + 89T^{2} \) |
| 97 | \( 1 - 0.343T + 97T^{2} \) |
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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
−8.997027504158089921985324218281, −8.225066809539496424417292086703, −7.72283621249452497011192608183, −6.70766403426222498241964632193, −5.59872840912335525457093924731, −5.10953837743043656398701519902, −4.19079668751028886155210833294, −3.41477777042949311280475784776, −2.37476320991162787116112382140, −0.62219726315800900212348935724,
0.62219726315800900212348935724, 2.37476320991162787116112382140, 3.41477777042949311280475784776, 4.19079668751028886155210833294, 5.10953837743043656398701519902, 5.59872840912335525457093924731, 6.70766403426222498241964632193, 7.72283621249452497011192608183, 8.225066809539496424417292086703, 8.997027504158089921985324218281
