| L(s) = 1 | − 2-s + 2.88·3-s + 4-s + 2.69·5-s − 2.88·6-s − 2.78·7-s − 8-s + 5.29·9-s − 2.69·10-s + 5.53·11-s + 2.88·12-s + 0.469·13-s + 2.78·14-s + 7.77·15-s + 16-s + 3.91·17-s − 5.29·18-s − 1.24·19-s + 2.69·20-s − 8.02·21-s − 5.53·22-s − 2.53·23-s − 2.88·24-s + 2.28·25-s − 0.469·26-s + 6.61·27-s − 2.78·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.66·3-s + 0.5·4-s + 1.20·5-s − 1.17·6-s − 1.05·7-s − 0.353·8-s + 1.76·9-s − 0.853·10-s + 1.66·11-s + 0.831·12-s + 0.130·13-s + 0.744·14-s + 2.00·15-s + 0.250·16-s + 0.949·17-s − 1.24·18-s − 0.284·19-s + 0.603·20-s − 1.75·21-s − 1.17·22-s − 0.528·23-s − 0.588·24-s + 0.457·25-s − 0.0920·26-s + 1.27·27-s − 0.526·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1502 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1502 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.723919195\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.723919195\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 751 | \( 1 - T \) |
| good | 3 | \( 1 - 2.88T + 3T^{2} \) |
| 5 | \( 1 - 2.69T + 5T^{2} \) |
| 7 | \( 1 + 2.78T + 7T^{2} \) |
| 11 | \( 1 - 5.53T + 11T^{2} \) |
| 13 | \( 1 - 0.469T + 13T^{2} \) |
| 17 | \( 1 - 3.91T + 17T^{2} \) |
| 19 | \( 1 + 1.24T + 19T^{2} \) |
| 23 | \( 1 + 2.53T + 23T^{2} \) |
| 29 | \( 1 - 0.404T + 29T^{2} \) |
| 31 | \( 1 + 4.00T + 31T^{2} \) |
| 37 | \( 1 + 6.81T + 37T^{2} \) |
| 41 | \( 1 + 3.40T + 41T^{2} \) |
| 43 | \( 1 + 8.33T + 43T^{2} \) |
| 47 | \( 1 - 9.56T + 47T^{2} \) |
| 53 | \( 1 - 8.42T + 53T^{2} \) |
| 59 | \( 1 - 2.27T + 59T^{2} \) |
| 61 | \( 1 + 1.43T + 61T^{2} \) |
| 67 | \( 1 - 4.13T + 67T^{2} \) |
| 71 | \( 1 + 7.24T + 71T^{2} \) |
| 73 | \( 1 - 6.23T + 73T^{2} \) |
| 79 | \( 1 - 3.12T + 79T^{2} \) |
| 83 | \( 1 + 0.603T + 83T^{2} \) |
| 89 | \( 1 - 10.9T + 89T^{2} \) |
| 97 | \( 1 - 6.63T + 97T^{2} \) |
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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
−9.348948783363726273920877092617, −8.945448089306977243998522545553, −8.171912714532183720104804926243, −7.08280305672974054194778223949, −6.54230272242490628308624101928, −5.62764995108710314259783854527, −3.88748652105890932410832683785, −3.29599932767458260162563267124, −2.20949357687253667242592105687, −1.41818963566063727865748206895,
1.41818963566063727865748206895, 2.20949357687253667242592105687, 3.29599932767458260162563267124, 3.88748652105890932410832683785, 5.62764995108710314259783854527, 6.54230272242490628308624101928, 7.08280305672974054194778223949, 8.171912714532183720104804926243, 8.945448089306977243998522545553, 9.348948783363726273920877092617
