L(s) = 1 | + 4i·7-s + 2i·13-s − 8·19-s − 4·31-s + 10i·37-s + 8i·43-s − 9·49-s + 14·61-s + 16i·67-s − 10i·73-s + 4·79-s − 8·91-s − 14i·97-s + 20i·103-s − 2·109-s + ⋯ |
L(s) = 1 | + 1.51i·7-s + 0.554i·13-s − 1.83·19-s − 0.718·31-s + 1.64i·37-s + 1.21i·43-s − 1.28·49-s + 1.79·61-s + 1.95i·67-s − 1.17i·73-s + 0.450·79-s − 0.838·91-s − 1.42i·97-s + 1.97i·103-s − 0.191·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.571010 + 0.923913i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.571010 + 0.923913i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( 1 - 4iT - 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 2iT - 13T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 + 8T + 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 - 10iT - 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 - 8iT - 43T^{2} \) |
| 47 | \( 1 - 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 14T + 61T^{2} \) |
| 67 | \( 1 - 16iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 10iT - 73T^{2} \) |
| 79 | \( 1 - 4T + 79T^{2} \) |
| 83 | \( 1 - 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 + 14iT - 97T^{2} \) |
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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
−10.34549328126821966547238549947, −9.408910698883420418039084080263, −8.702397642281400021030677448643, −8.114318136365360461235917057505, −6.77232627370478689111305337850, −6.11075549340203458669385944203, −5.16675416129077266926274467116, −4.16912952229599195397693272089, −2.80506258313838782912658044500, −1.86995435805798286223267580506,
0.49716465820202011748401020942, 2.09726352036911341379616119879, 3.63459964505567699601710174536, 4.27205744345434838740889966837, 5.43364868088387971316187533598, 6.54957151802260648839986603555, 7.26807289819363886526191691576, 8.083068790909649331171553212913, 8.975419529647267063734345394210, 10.04029464851918290155355129228