L(s) = 1 | + i·2-s − 4-s + i·7-s − i·8-s + 5·11-s − 6i·13-s − 14-s + 16-s − i·17-s + 3·19-s + 5i·22-s + 6·26-s − i·28-s − 6·29-s − 4·31-s + i·32-s + ⋯ |
L(s) = 1 | + 0.707i·2-s − 0.5·4-s + 0.377i·7-s − 0.353i·8-s + 1.50·11-s − 1.66i·13-s − 0.267·14-s + 0.250·16-s − 0.242i·17-s + 0.688·19-s + 1.06i·22-s + 1.17·26-s − 0.188i·28-s − 1.11·29-s − 0.718·31-s + 0.176i·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.567479516\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.567479516\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - iT \) |
good | 11 | \( 1 - 5T + 11T^{2} \) |
| 13 | \( 1 + 6iT - 13T^{2} \) |
| 17 | \( 1 + iT - 17T^{2} \) |
| 19 | \( 1 - 3T + 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 + 8iT - 37T^{2} \) |
| 41 | \( 1 + 11T + 41T^{2} \) |
| 43 | \( 1 + 8iT - 43T^{2} \) |
| 47 | \( 1 - 2iT - 47T^{2} \) |
| 53 | \( 1 + 4iT - 53T^{2} \) |
| 59 | \( 1 - 4T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 + 9iT - 67T^{2} \) |
| 71 | \( 1 - 10T + 71T^{2} \) |
| 73 | \( 1 + 7iT - 73T^{2} \) |
| 79 | \( 1 - 2T + 79T^{2} \) |
| 83 | \( 1 + 11iT - 83T^{2} \) |
| 89 | \( 1 + 11T + 89T^{2} \) |
| 97 | \( 1 - 10iT - 97T^{2} \) |
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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
−8.647550035908286158871184300160, −7.76603679378902413954779017383, −7.18559469412112112694076262632, −6.35628908001268040720223029535, −5.56940118472467535869881663402, −5.08876612674051165008006426965, −3.82591323704486498966740123322, −3.29831504988979286938665182240, −1.85702953716993131741810715062, −0.52581178960635267967458547310,
1.26151310040410465549325609124, 1.89161967619458329935438229526, 3.27044647177164368629895254730, 3.97859582970042917261455859819, 4.57512033006114539987923451965, 5.63126177784664005361074925686, 6.67356107790910299249991892052, 7.04153297606879144723029512768, 8.193752853026428756146186289973, 8.940999227549445112879363638634