| L(s) = 1 | + 2-s − 3-s + 4-s + 3.35·5-s − 6-s + 2.89·7-s + 8-s + 9-s + 3.35·10-s + 2.23·11-s − 12-s + 13-s + 2.89·14-s − 3.35·15-s + 16-s − 3.39·17-s + 18-s + 7.31·19-s + 3.35·20-s − 2.89·21-s + 2.23·22-s − 3.61·23-s − 24-s + 6.25·25-s + 26-s − 27-s + 2.89·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s + 1.50·5-s − 0.408·6-s + 1.09·7-s + 0.353·8-s + 0.333·9-s + 1.06·10-s + 0.673·11-s − 0.288·12-s + 0.277·13-s + 0.773·14-s − 0.866·15-s + 0.250·16-s − 0.823·17-s + 0.235·18-s + 1.67·19-s + 0.750·20-s − 0.631·21-s + 0.476·22-s − 0.753·23-s − 0.204·24-s + 1.25·25-s + 0.196·26-s − 0.192·27-s + 0.547·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.892431457\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.892431457\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 103 | \( 1 - T \) |
| good | 5 | \( 1 - 3.35T + 5T^{2} \) |
| 7 | \( 1 - 2.89T + 7T^{2} \) |
| 11 | \( 1 - 2.23T + 11T^{2} \) |
| 17 | \( 1 + 3.39T + 17T^{2} \) |
| 19 | \( 1 - 7.31T + 19T^{2} \) |
| 23 | \( 1 + 3.61T + 23T^{2} \) |
| 29 | \( 1 + 4.47T + 29T^{2} \) |
| 31 | \( 1 - 7.31T + 31T^{2} \) |
| 37 | \( 1 - 0.856T + 37T^{2} \) |
| 41 | \( 1 + 6.78T + 41T^{2} \) |
| 43 | \( 1 - 8.47T + 43T^{2} \) |
| 47 | \( 1 + 3.31T + 47T^{2} \) |
| 53 | \( 1 - 0.133T + 53T^{2} \) |
| 59 | \( 1 + 4.10T + 59T^{2} \) |
| 61 | \( 1 - 11.3T + 61T^{2} \) |
| 67 | \( 1 + 6.04T + 67T^{2} \) |
| 71 | \( 1 + 7.81T + 71T^{2} \) |
| 73 | \( 1 - 3.56T + 73T^{2} \) |
| 79 | \( 1 - 10.9T + 79T^{2} \) |
| 83 | \( 1 + 12.9T + 83T^{2} \) |
| 89 | \( 1 - 0.160T + 89T^{2} \) |
| 97 | \( 1 - 8.23T + 97T^{2} \) |
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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
−7.64770683600603979231061281644, −6.90581897814838518044097919549, −6.19964722395926189001779708569, −5.75012789572277254983355466564, −5.08278775029745173329793641603, −4.55376476998790560267329998303, −3.63902103026400457985763626350, −2.53639366996488752646199594734, −1.74960552904095600161094708289, −1.14028872540681543323075013889,
1.14028872540681543323075013889, 1.74960552904095600161094708289, 2.53639366996488752646199594734, 3.63902103026400457985763626350, 4.55376476998790560267329998303, 5.08278775029745173329793641603, 5.75012789572277254983355466564, 6.19964722395926189001779708569, 6.90581897814838518044097919549, 7.64770683600603979231061281644
