L(s) = 1 | + (1.82 − 1.82i)2-s + (1.43 − 0.964i)3-s − 4.66i·4-s + (−0.624 + 0.624i)5-s + (0.865 − 4.38i)6-s + (−1.18 + 1.18i)7-s + (−4.86 − 4.86i)8-s + (1.13 − 2.77i)9-s + 2.28i·10-s + (0.253 + 0.253i)11-s + (−4.49 − 6.70i)12-s + 4.34i·14-s + (−0.296 + 1.50i)15-s − 8.41·16-s − 2.62·17-s + (−2.98 − 7.14i)18-s + ⋯ |
L(s) = 1 | + (1.29 − 1.29i)2-s + (0.830 − 0.556i)3-s − 2.33i·4-s + (−0.279 + 0.279i)5-s + (0.353 − 1.79i)6-s + (−0.449 + 0.449i)7-s + (−1.71 − 1.71i)8-s + (0.379 − 0.925i)9-s + 0.721i·10-s + (0.0763 + 0.0763i)11-s + (−1.29 − 1.93i)12-s + 1.15i·14-s + (−0.0764 + 0.387i)15-s − 2.10·16-s − 0.637·17-s + (−0.703 − 1.68i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.755 + 0.654i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.755 + 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.13229 - 3.03594i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.13229 - 3.03594i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-1.43 + 0.964i)T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + (-1.82 + 1.82i)T - 2iT^{2} \) |
| 5 | \( 1 + (0.624 - 0.624i)T - 5iT^{2} \) |
| 7 | \( 1 + (1.18 - 1.18i)T - 7iT^{2} \) |
| 11 | \( 1 + (-0.253 - 0.253i)T + 11iT^{2} \) |
| 17 | \( 1 + 2.62T + 17T^{2} \) |
| 19 | \( 1 + (-4.32 - 4.32i)T + 19iT^{2} \) |
| 23 | \( 1 - 5.41T + 23T^{2} \) |
| 29 | \( 1 + 8.37iT - 29T^{2} \) |
| 31 | \( 1 + (-1.27 - 1.27i)T + 31iT^{2} \) |
| 37 | \( 1 + (2.44 - 2.44i)T - 37iT^{2} \) |
| 41 | \( 1 + (4.89 - 4.89i)T - 41iT^{2} \) |
| 43 | \( 1 + 0.952iT - 43T^{2} \) |
| 47 | \( 1 + (-5.33 - 5.33i)T + 47iT^{2} \) |
| 53 | \( 1 - 9.69iT - 53T^{2} \) |
| 59 | \( 1 + (7.28 + 7.28i)T + 59iT^{2} \) |
| 61 | \( 1 - 4.87T + 61T^{2} \) |
| 67 | \( 1 + (8.90 + 8.90i)T + 67iT^{2} \) |
| 71 | \( 1 + (-2.27 + 2.27i)T - 71iT^{2} \) |
| 73 | \( 1 + (-0.246 + 0.246i)T - 73iT^{2} \) |
| 79 | \( 1 + 3.68T + 79T^{2} \) |
| 83 | \( 1 + (8.47 - 8.47i)T - 83iT^{2} \) |
| 89 | \( 1 + (4.84 + 4.84i)T + 89iT^{2} \) |
| 97 | \( 1 + (-3.74 - 3.74i)T + 97iT^{2} \) |
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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
−10.85677144220143632920764616924, −9.779140268385654499158422620857, −9.166468530540983516471859736686, −7.81252209250052693418818280858, −6.64813333692663153679446773544, −5.72066894966160769967284672295, −4.42777798350135552997551140529, −3.36253604899361020433913534345, −2.73397216952312935597594290666, −1.44558831448784815116640318700,
2.88267553291978845141445332099, 3.74653030804431720266907372715, 4.66416534965008149886770951087, 5.39450290632457885135899148820, 6.90618909340177840440181245631, 7.24690450258932346954649367331, 8.486725889141171506466795213340, 9.005862325778903000227954528277, 10.31067786541926110466154056571, 11.46891878855783666256243464940