L(s) = 1 | + 0.414i·2-s + 1.82·4-s + 4.82i·7-s + 1.58i·8-s + 11-s + 5.65i·13-s − 1.99·14-s + 3·16-s − 6.82i·17-s + 1.17·19-s + 0.414i·22-s + 4i·23-s − 2.34·26-s + 8.82i·28-s + 0.828·29-s + ⋯ |
L(s) = 1 | + 0.292i·2-s + 0.914·4-s + 1.82i·7-s + 0.560i·8-s + 0.301·11-s + 1.56i·13-s − 0.534·14-s + 0.750·16-s − 1.65i·17-s + 0.268·19-s + 0.0883i·22-s + 0.834i·23-s − 0.459·26-s + 1.66i·28-s + 0.153·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.227335406\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.227335406\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 - T \) |
good | 2 | \( 1 - 0.414iT - 2T^{2} \) |
| 7 | \( 1 - 4.82iT - 7T^{2} \) |
| 13 | \( 1 - 5.65iT - 13T^{2} \) |
| 17 | \( 1 + 6.82iT - 17T^{2} \) |
| 19 | \( 1 - 1.17T + 19T^{2} \) |
| 23 | \( 1 - 4iT - 23T^{2} \) |
| 29 | \( 1 - 0.828T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 0.343iT - 37T^{2} \) |
| 41 | \( 1 - 0.828T + 41T^{2} \) |
| 43 | \( 1 + 3.17iT - 43T^{2} \) |
| 47 | \( 1 + 4iT - 47T^{2} \) |
| 53 | \( 1 - 13.3iT - 53T^{2} \) |
| 59 | \( 1 + 4T + 59T^{2} \) |
| 61 | \( 1 + 0.343T + 61T^{2} \) |
| 67 | \( 1 + 5.65iT - 67T^{2} \) |
| 71 | \( 1 + 13.6T + 71T^{2} \) |
| 73 | \( 1 + 11.3iT - 73T^{2} \) |
| 79 | \( 1 - 8.48T + 79T^{2} \) |
| 83 | \( 1 - 10iT - 83T^{2} \) |
| 89 | \( 1 + 7.65T + 89T^{2} \) |
| 97 | \( 1 + 0.343iT - 97T^{2} \) |
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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
−9.293320117457178538631982482247, −8.470684269732932870276020766259, −7.48523824736595798064823709875, −6.86741632614896323544157588467, −6.08661421270061694517609354089, −5.45866781020379517862430150697, −4.62629272554037818486928924440, −3.20794363665649943270852398275, −2.44135873925448570708139591146, −1.65623609262533898635247304361,
0.72191048936972842072221422669, 1.62503927732425910002157252449, 2.99461009951280061993440136305, 3.68486096630303576990780475333, 4.46900875007296364615234843844, 5.73112847783807713090698179165, 6.46342116206719107276429639516, 7.16583634226212540534802536721, 7.86756777053947113082711152536, 8.412356141346502090162147934824