L(s) = 1 | + i·2-s − 2.96i·3-s − 4-s + 2.96·6-s + 1.83i·7-s − i·8-s − 5.79·9-s − 1.83·11-s + 2.96i·12-s + 2.41i·13-s − 1.83·14-s + 16-s + 2.78i·17-s − 5.79i·18-s + 1.67·19-s + ⋯ |
L(s) = 1 | + 0.707i·2-s − 1.71i·3-s − 0.5·4-s + 1.21·6-s + 0.692i·7-s − 0.353i·8-s − 1.93·9-s − 0.552·11-s + 0.856i·12-s + 0.670i·13-s − 0.489·14-s + 0.250·16-s + 0.675i·17-s − 1.36i·18-s + 0.383·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1250 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1250 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8388660713\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8388660713\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - iT \) |
| 5 | \( 1 \) |
good | 3 | \( 1 + 2.96iT - 3T^{2} \) |
| 7 | \( 1 - 1.83iT - 7T^{2} \) |
| 11 | \( 1 + 1.83T + 11T^{2} \) |
| 13 | \( 1 - 2.41iT - 13T^{2} \) |
| 17 | \( 1 - 2.78iT - 17T^{2} \) |
| 19 | \( 1 - 1.67T + 19T^{2} \) |
| 23 | \( 1 - 7.76iT - 23T^{2} \) |
| 29 | \( 1 + 7.58T + 29T^{2} \) |
| 31 | \( 1 + 5.29T + 31T^{2} \) |
| 37 | \( 1 + 1.31iT - 37T^{2} \) |
| 41 | \( 1 - 3.51T + 41T^{2} \) |
| 43 | \( 1 - 4.30iT - 43T^{2} \) |
| 47 | \( 1 - 1.83iT - 47T^{2} \) |
| 53 | \( 1 - 6.51iT - 53T^{2} \) |
| 59 | \( 1 - 9.06T + 59T^{2} \) |
| 61 | \( 1 - 2.58T + 61T^{2} \) |
| 67 | \( 1 - 9.50iT - 67T^{2} \) |
| 71 | \( 1 - 0.305T + 71T^{2} \) |
| 73 | \( 1 + 14.9iT - 73T^{2} \) |
| 79 | \( 1 - 3.46T + 79T^{2} \) |
| 83 | \( 1 - 6.37iT - 83T^{2} \) |
| 89 | \( 1 - 3.32T + 89T^{2} \) |
| 97 | \( 1 - 11.0iT - 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.466963597464925185934220393701, −8.908044444320560859532602920847, −7.88707030709441928906547585371, −7.52219638466885184364000152539, −6.71038170258528781460770362355, −5.80676652747203406981083604678, −5.38563825396175205026475998952, −3.75727419712785892919034017607, −2.41502187586504972719759957706, −1.42686658328713912079255982451,
0.35839607037691932125531672962, 2.48715653810262041909912497459, 3.46920833661267858839790320077, 4.14559232662747553593798641940, 5.04193299134563080051864341503, 5.63263049601691481806075423309, 7.14109496137102645565098015401, 8.198254606157045534592550748266, 8.994632709290590624955956805440, 9.764065146135516144399318650758