| L(s) = 1 | + i·2-s − 4-s − i·8-s − 3i·11-s + 3.46i·13-s + 16-s + 3·22-s − 5·25-s − 3.46·26-s + 9i·29-s − 1.73i·31-s + i·32-s − 8·37-s + 10.3·41-s + 4·43-s + 3i·44-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.5·4-s − 0.353i·8-s − 0.904i·11-s + 0.960i·13-s + 0.250·16-s + 0.639·22-s − 25-s − 0.679·26-s + 1.67i·29-s − 0.311i·31-s + 0.176i·32-s − 1.31·37-s + 1.62·41-s + 0.609·43-s + 0.452i·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.654 - 0.755i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.654 - 0.755i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.223273165\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.223273165\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - iT \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 5T^{2} \) |
| 11 | \( 1 + 3iT - 11T^{2} \) |
| 13 | \( 1 - 3.46iT - 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 - 9iT - 29T^{2} \) |
| 31 | \( 1 + 1.73iT - 31T^{2} \) |
| 37 | \( 1 + 8T + 37T^{2} \) |
| 41 | \( 1 - 10.3T + 41T^{2} \) |
| 43 | \( 1 - 4T + 43T^{2} \) |
| 47 | \( 1 - 10.3T + 47T^{2} \) |
| 53 | \( 1 - 6iT - 53T^{2} \) |
| 59 | \( 1 - 5.19T + 59T^{2} \) |
| 61 | \( 1 - 13.8iT - 61T^{2} \) |
| 67 | \( 1 - 2T + 67T^{2} \) |
| 71 | \( 1 - 12iT - 71T^{2} \) |
| 73 | \( 1 - 5.19iT - 73T^{2} \) |
| 79 | \( 1 + 13T + 79T^{2} \) |
| 83 | \( 1 + 5.19T + 83T^{2} \) |
| 89 | \( 1 + 10.3T + 89T^{2} \) |
| 97 | \( 1 + 8.66iT - 97T^{2} \) |
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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
−8.871580380558174908287251089072, −8.533299262418267129049752013481, −7.40446607665820511407530553811, −7.00883344249636638823367877524, −5.93547577587174152902369081047, −5.54243244906231032844768619512, −4.38317418358235772484216556305, −3.73850560760247235730777413811, −2.56096117314947496579523076061, −1.17415559689160526998408073769,
0.43902315514239629342500150671, 1.84395497427287638255664399873, 2.66734810818300502285849989552, 3.73739690963575306371655323228, 4.45494770468951284887059641161, 5.40148200973004817018802523135, 6.11027004683910583175952793596, 7.26420727806180690533881199717, 7.86006074265710689335683392557, 8.653419652234770887780091947343
