| L(s) = 1 | + 2-s − 2.44·3-s + 4-s − 2.44·6-s + 4.44·7-s + 8-s + 2.99·9-s + 2·11-s − 2.44·12-s − 2.44·13-s + 4.44·14-s + 16-s − 2·17-s + 2.99·18-s + 6.44·19-s − 10.8·21-s + 2·22-s − 2.44·23-s − 2.44·24-s − 2.44·26-s + 4.44·28-s − 29-s − 3·31-s + 32-s − 4.89·33-s − 2·34-s + 2.99·36-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.41·3-s + 0.5·4-s − 0.999·6-s + 1.68·7-s + 0.353·8-s + 0.999·9-s + 0.603·11-s − 0.707·12-s − 0.679·13-s + 1.18·14-s + 0.250·16-s − 0.485·17-s + 0.707·18-s + 1.47·19-s − 2.37·21-s + 0.426·22-s − 0.510·23-s − 0.499·24-s − 0.480·26-s + 0.840·28-s − 0.185·29-s − 0.538·31-s + 0.176·32-s − 0.852·33-s − 0.342·34-s + 0.499·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.018124112\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.018124112\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 29 | \( 1 + T \) |
| good | 3 | \( 1 + 2.44T + 3T^{2} \) |
| 7 | \( 1 - 4.44T + 7T^{2} \) |
| 11 | \( 1 - 2T + 11T^{2} \) |
| 13 | \( 1 + 2.44T + 13T^{2} \) |
| 17 | \( 1 + 2T + 17T^{2} \) |
| 19 | \( 1 - 6.44T + 19T^{2} \) |
| 23 | \( 1 + 2.44T + 23T^{2} \) |
| 31 | \( 1 + 3T + 31T^{2} \) |
| 37 | \( 1 + 3.44T + 37T^{2} \) |
| 41 | \( 1 - 11.3T + 41T^{2} \) |
| 43 | \( 1 + 8.89T + 43T^{2} \) |
| 47 | \( 1 - 5.89T + 47T^{2} \) |
| 53 | \( 1 - 10.4T + 53T^{2} \) |
| 59 | \( 1 - 8.55T + 59T^{2} \) |
| 61 | \( 1 - 3.44T + 61T^{2} \) |
| 67 | \( 1 + 13.4T + 67T^{2} \) |
| 71 | \( 1 - 11.3T + 71T^{2} \) |
| 73 | \( 1 - 13.3T + 73T^{2} \) |
| 79 | \( 1 + 2.89T + 79T^{2} \) |
| 83 | \( 1 + 6T + 83T^{2} \) |
| 89 | \( 1 - 3.55T + 89T^{2} \) |
| 97 | \( 1 - 7.34T + 97T^{2} \) |
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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
−9.758492056733616712706281303483, −8.635156912135106391581791532783, −7.56827793228039758588157569749, −7.01913348023082375639211154063, −5.93488349961422357683220339908, −5.27426545444532335896337217715, −4.76572845175765916903867323079, −3.86151474800745899831388840079, −2.21228206811145328192604715693, −1.05515224977993121655757897581,
1.05515224977993121655757897581, 2.21228206811145328192604715693, 3.86151474800745899831388840079, 4.76572845175765916903867323079, 5.27426545444532335896337217715, 5.93488349961422357683220339908, 7.01913348023082375639211154063, 7.56827793228039758588157569749, 8.635156912135106391581791532783, 9.758492056733616712706281303483
