L(s) = 1 | + 2-s + 3-s + 4-s + 0.585·5-s + 6-s + 8-s + 9-s + 0.585·10-s − 0.414·11-s + 12-s − 13-s + 0.585·15-s + 16-s + 2.41·17-s + 18-s + 2.17·19-s + 0.585·20-s − 0.414·22-s + 1.41·23-s + 24-s − 4.65·25-s − 26-s + 27-s − 1.82·29-s + 0.585·30-s + 8.48·31-s + 32-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.261·5-s + 0.408·6-s + 0.353·8-s + 0.333·9-s + 0.185·10-s − 0.124·11-s + 0.288·12-s − 0.277·13-s + 0.151·15-s + 0.250·16-s + 0.585·17-s + 0.235·18-s + 0.498·19-s + 0.130·20-s − 0.0883·22-s + 0.294·23-s + 0.204·24-s − 0.931·25-s − 0.196·26-s + 0.192·27-s − 0.339·29-s + 0.106·30-s + 1.52·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3822 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3822 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.174381773\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.174381773\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 \) |
| 13 | \( 1 + T \) |
good | 5 | \( 1 - 0.585T + 5T^{2} \) |
| 11 | \( 1 + 0.414T + 11T^{2} \) |
| 17 | \( 1 - 2.41T + 17T^{2} \) |
| 19 | \( 1 - 2.17T + 19T^{2} \) |
| 23 | \( 1 - 1.41T + 23T^{2} \) |
| 29 | \( 1 + 1.82T + 29T^{2} \) |
| 31 | \( 1 - 8.48T + 31T^{2} \) |
| 37 | \( 1 - 1.41T + 37T^{2} \) |
| 41 | \( 1 - 9.89T + 41T^{2} \) |
| 43 | \( 1 - 6.48T + 43T^{2} \) |
| 47 | \( 1 + T + 47T^{2} \) |
| 53 | \( 1 - 9.48T + 53T^{2} \) |
| 59 | \( 1 + 2.07T + 59T^{2} \) |
| 61 | \( 1 - 4.41T + 61T^{2} \) |
| 67 | \( 1 - 1.82T + 67T^{2} \) |
| 71 | \( 1 + 5T + 71T^{2} \) |
| 73 | \( 1 + 1.41T + 73T^{2} \) |
| 79 | \( 1 + 11.6T + 79T^{2} \) |
| 83 | \( 1 + 7.65T + 83T^{2} \) |
| 89 | \( 1 + 2.58T + 89T^{2} \) |
| 97 | \( 1 - 0.928T + 97T^{2} \) |
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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
−8.374147696035896582432613341195, −7.64230264662072331506549624389, −7.11807322484073014623242411570, −6.08934666023661149400415239502, −5.55098167671041131397415648604, −4.59249532232073551331564012937, −3.90133318381047795293898974819, −2.94466706443188630048931573478, −2.30269776782336020683090928412, −1.09755613866173283200745994117,
1.09755613866173283200745994117, 2.30269776782336020683090928412, 2.94466706443188630048931573478, 3.90133318381047795293898974819, 4.59249532232073551331564012937, 5.55098167671041131397415648604, 6.08934666023661149400415239502, 7.11807322484073014623242411570, 7.64230264662072331506549624389, 8.374147696035896582432613341195