L(s) = 1 | + (−0.707 − 0.707i)3-s + (−1.65 + 1.50i)5-s − 2.58·7-s + 1.00i·9-s + (4.39 + 4.39i)11-s + (−0.417 − 0.417i)13-s + (2.23 + 0.110i)15-s − 4.40i·17-s + (−4.53 + 4.53i)19-s + (1.83 + 1.83i)21-s − 0.281·23-s + (0.494 − 4.97i)25-s + (0.707 − 0.707i)27-s + (−3.73 + 3.73i)29-s + 3.05·31-s + ⋯ |
L(s) = 1 | + (−0.408 − 0.408i)3-s + (−0.741 + 0.671i)5-s − 0.978·7-s + 0.333i·9-s + (1.32 + 1.32i)11-s + (−0.115 − 0.115i)13-s + (0.576 + 0.0286i)15-s − 1.06i·17-s + (−1.04 + 1.04i)19-s + (0.399 + 0.399i)21-s − 0.0586·23-s + (0.0989 − 0.995i)25-s + (0.136 − 0.136i)27-s + (−0.693 + 0.693i)29-s + 0.548·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.782 + 0.622i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.782 + 0.622i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.1555196712\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1555196712\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.707 + 0.707i)T \) |
| 5 | \( 1 + (1.65 - 1.50i)T \) |
good | 7 | \( 1 + 2.58T + 7T^{2} \) |
| 11 | \( 1 + (-4.39 - 4.39i)T + 11iT^{2} \) |
| 13 | \( 1 + (0.417 + 0.417i)T + 13iT^{2} \) |
| 17 | \( 1 + 4.40iT - 17T^{2} \) |
| 19 | \( 1 + (4.53 - 4.53i)T - 19iT^{2} \) |
| 23 | \( 1 + 0.281T + 23T^{2} \) |
| 29 | \( 1 + (3.73 - 3.73i)T - 29iT^{2} \) |
| 31 | \( 1 - 3.05T + 31T^{2} \) |
| 37 | \( 1 + (-5.26 + 5.26i)T - 37iT^{2} \) |
| 41 | \( 1 + 5.16iT - 41T^{2} \) |
| 43 | \( 1 + (2.66 - 2.66i)T - 43iT^{2} \) |
| 47 | \( 1 - 7.45iT - 47T^{2} \) |
| 53 | \( 1 + (-2.89 + 2.89i)T - 53iT^{2} \) |
| 59 | \( 1 + (4.60 + 4.60i)T + 59iT^{2} \) |
| 61 | \( 1 + (-0.211 + 0.211i)T - 61iT^{2} \) |
| 67 | \( 1 + (7.17 + 7.17i)T + 67iT^{2} \) |
| 71 | \( 1 + 15.9iT - 71T^{2} \) |
| 73 | \( 1 + 10.5T + 73T^{2} \) |
| 79 | \( 1 + 4.53T + 79T^{2} \) |
| 83 | \( 1 + (4.56 + 4.56i)T + 83iT^{2} \) |
| 89 | \( 1 + 10.2iT - 89T^{2} \) |
| 97 | \( 1 - 6.78iT - 97T^{2} \) |
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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
−9.033833188978553770774642583314, −7.82941278222147905498437022590, −7.21289682771737214146571942047, −6.59987185361543250196323142664, −6.01687078274811896641890787362, −4.64512043154572658349419786848, −3.92885421980546508984594603464, −2.95013074278003176722474717260, −1.75217562202586395807913144032, −0.06776553153086998351816459955,
1.14087019353839175185729776999, 2.95483114255328982732353891780, 3.94794320982671561453825904988, 4.32245084654209572600563102544, 5.62649053162380067444079852297, 6.31037981189578011905208194352, 6.92585928829282015911993170164, 8.243377055085991452622150603825, 8.731473646156673316580782586851, 9.404734843423692796212048774440