| L(s) = 1 | + 2-s + 3.36·3-s + 4-s − 5-s + 3.36·6-s − 3.18·7-s + 8-s + 8.33·9-s − 10-s + 1.51·11-s + 3.36·12-s + 0.765·13-s − 3.18·14-s − 3.36·15-s + 16-s − 4.45·17-s + 8.33·18-s − 0.610·19-s − 20-s − 10.7·21-s + 1.51·22-s + 6.02·23-s + 3.36·24-s + 25-s + 0.765·26-s + 17.9·27-s − 3.18·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.94·3-s + 0.5·4-s − 0.447·5-s + 1.37·6-s − 1.20·7-s + 0.353·8-s + 2.77·9-s − 0.316·10-s + 0.455·11-s + 0.971·12-s + 0.212·13-s − 0.850·14-s − 0.869·15-s + 0.250·16-s − 1.08·17-s + 1.96·18-s − 0.139·19-s − 0.223·20-s − 2.33·21-s + 0.321·22-s + 1.25·23-s + 0.687·24-s + 0.200·25-s + 0.150·26-s + 3.45·27-s − 0.601·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.016250575\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.016250575\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 601 | \( 1 + T \) |
| good | 3 | \( 1 - 3.36T + 3T^{2} \) |
| 7 | \( 1 + 3.18T + 7T^{2} \) |
| 11 | \( 1 - 1.51T + 11T^{2} \) |
| 13 | \( 1 - 0.765T + 13T^{2} \) |
| 17 | \( 1 + 4.45T + 17T^{2} \) |
| 19 | \( 1 + 0.610T + 19T^{2} \) |
| 23 | \( 1 - 6.02T + 23T^{2} \) |
| 29 | \( 1 - 8.30T + 29T^{2} \) |
| 31 | \( 1 + 2.60T + 31T^{2} \) |
| 37 | \( 1 - 8.12T + 37T^{2} \) |
| 41 | \( 1 - 5.08T + 41T^{2} \) |
| 43 | \( 1 + 2.76T + 43T^{2} \) |
| 47 | \( 1 - 10.1T + 47T^{2} \) |
| 53 | \( 1 + 12.2T + 53T^{2} \) |
| 59 | \( 1 + 10.0T + 59T^{2} \) |
| 61 | \( 1 - 13.5T + 61T^{2} \) |
| 67 | \( 1 - 4.73T + 67T^{2} \) |
| 71 | \( 1 + 0.106T + 71T^{2} \) |
| 73 | \( 1 + 1.08T + 73T^{2} \) |
| 79 | \( 1 + 6.90T + 79T^{2} \) |
| 83 | \( 1 - 13.1T + 83T^{2} \) |
| 89 | \( 1 + 12.3T + 89T^{2} \) |
| 97 | \( 1 + 3.16T + 97T^{2} \) |
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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
−8.096674847478573611577905506429, −7.32910779757830025140885014528, −6.75212607945580762274685801902, −6.24588580761787411326673560593, −4.78074599913097221263951117435, −4.19595485507485373310622874010, −3.54174195381854987291637620411, −2.89703838033683291283880488265, −2.37645233259584679004220549749, −1.11296614029614949099401299038,
1.11296614029614949099401299038, 2.37645233259584679004220549749, 2.89703838033683291283880488265, 3.54174195381854987291637620411, 4.19595485507485373310622874010, 4.78074599913097221263951117435, 6.24588580761787411326673560593, 6.75212607945580762274685801902, 7.32910779757830025140885014528, 8.096674847478573611577905506429
