| L(s) = 1 | − 2-s − 2.04·3-s + 4-s + 4.28·5-s + 2.04·6-s + 0.891·7-s − 8-s + 1.17·9-s − 4.28·10-s + 0.374·11-s − 2.04·12-s + 6.96·13-s − 0.891·14-s − 8.75·15-s + 16-s + 5.42·17-s − 1.17·18-s − 3.90·19-s + 4.28·20-s − 1.82·21-s − 0.374·22-s − 23-s + 2.04·24-s + 13.3·25-s − 6.96·26-s + 3.73·27-s + 0.891·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.17·3-s + 0.5·4-s + 1.91·5-s + 0.833·6-s + 0.336·7-s − 0.353·8-s + 0.390·9-s − 1.35·10-s + 0.112·11-s − 0.589·12-s + 1.93·13-s − 0.238·14-s − 2.25·15-s + 0.250·16-s + 1.31·17-s − 0.276·18-s − 0.896·19-s + 0.958·20-s − 0.397·21-s − 0.0798·22-s − 0.208·23-s + 0.416·24-s + 2.67·25-s − 1.36·26-s + 0.718·27-s + 0.168·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1334 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1334 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.321999017\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.321999017\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 - T \) |
| good | 3 | \( 1 + 2.04T + 3T^{2} \) |
| 5 | \( 1 - 4.28T + 5T^{2} \) |
| 7 | \( 1 - 0.891T + 7T^{2} \) |
| 11 | \( 1 - 0.374T + 11T^{2} \) |
| 13 | \( 1 - 6.96T + 13T^{2} \) |
| 17 | \( 1 - 5.42T + 17T^{2} \) |
| 19 | \( 1 + 3.90T + 19T^{2} \) |
| 31 | \( 1 - 2.37T + 31T^{2} \) |
| 37 | \( 1 + 5.60T + 37T^{2} \) |
| 41 | \( 1 + 7.60T + 41T^{2} \) |
| 43 | \( 1 + 4.45T + 43T^{2} \) |
| 47 | \( 1 + 5.56T + 47T^{2} \) |
| 53 | \( 1 + 3.94T + 53T^{2} \) |
| 59 | \( 1 - 9.38T + 59T^{2} \) |
| 61 | \( 1 - 13.5T + 61T^{2} \) |
| 67 | \( 1 + 8.70T + 67T^{2} \) |
| 71 | \( 1 - 10.4T + 71T^{2} \) |
| 73 | \( 1 + 10.5T + 73T^{2} \) |
| 79 | \( 1 - 15.0T + 79T^{2} \) |
| 83 | \( 1 + 16.1T + 83T^{2} \) |
| 89 | \( 1 + 0.519T + 89T^{2} \) |
| 97 | \( 1 - 0.700T + 97T^{2} \) |
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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
−9.981041714326055793230103984339, −8.758701114997745434742031926947, −8.346616112119115941996580954285, −6.76254339558163528187189845680, −6.27974975179131384957396197072, −5.69526659661794130768485195636, −4.99583517804399753129486586771, −3.32509732493313467973532903195, −1.85638528208037249683311629996, −1.09545359167212322957761179958,
1.09545359167212322957761179958, 1.85638528208037249683311629996, 3.32509732493313467973532903195, 4.99583517804399753129486586771, 5.69526659661794130768485195636, 6.27974975179131384957396197072, 6.76254339558163528187189845680, 8.346616112119115941996580954285, 8.758701114997745434742031926947, 9.981041714326055793230103984339
