L(s) = 1 | + (0.707 + 0.707i)2-s + 1.00i·4-s + (0.489 + 2.18i)5-s + (0.707 − 0.707i)7-s + (−0.707 + 0.707i)8-s + (−1.19 + 1.88i)10-s + 1.77i·11-s + (0.692 + 0.692i)13-s + 1.00·14-s − 1.00·16-s + (2.39 + 2.39i)17-s + (−2.18 + 0.489i)20-s + (−1.25 + 1.25i)22-s + (−3.38 + 3.38i)23-s + (−4.52 + 2.13i)25-s + 0.979i·26-s + ⋯ |
L(s) = 1 | + (0.499 + 0.499i)2-s + 0.500i·4-s + (0.218 + 0.975i)5-s + (0.267 − 0.267i)7-s + (−0.250 + 0.250i)8-s + (−0.378 + 0.597i)10-s + 0.536i·11-s + (0.192 + 0.192i)13-s + 0.267·14-s − 0.250·16-s + (0.580 + 0.580i)17-s + (−0.487 + 0.109i)20-s + (−0.268 + 0.268i)22-s + (−0.705 + 0.705i)23-s + (−0.904 + 0.427i)25-s + 0.192i·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.278 - 0.960i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.278 - 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.17373 + 1.56306i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.17373 + 1.56306i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.707 - 0.707i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-0.489 - 2.18i)T \) |
| 7 | \( 1 + (-0.707 + 0.707i)T \) |
good | 11 | \( 1 - 1.77iT - 11T^{2} \) |
| 13 | \( 1 + (-0.692 - 0.692i)T + 13iT^{2} \) |
| 17 | \( 1 + (-2.39 - 2.39i)T + 17iT^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 23 | \( 1 + (3.38 - 3.38i)T - 23iT^{2} \) |
| 29 | \( 1 - 4.42T + 29T^{2} \) |
| 31 | \( 1 + 5.77T + 31T^{2} \) |
| 37 | \( 1 + (-5.91 + 5.91i)T - 37iT^{2} \) |
| 41 | \( 1 + 0.807iT - 41T^{2} \) |
| 43 | \( 1 + (4.64 + 4.64i)T + 43iT^{2} \) |
| 47 | \( 1 + (-7.47 - 7.47i)T + 47iT^{2} \) |
| 53 | \( 1 + (3.56 - 3.56i)T - 53iT^{2} \) |
| 59 | \( 1 - 5.89T + 59T^{2} \) |
| 61 | \( 1 - 2.39T + 61T^{2} \) |
| 67 | \( 1 + (0.641 - 0.641i)T - 67iT^{2} \) |
| 71 | \( 1 + 10.6iT - 71T^{2} \) |
| 73 | \( 1 + (-5.70 - 5.70i)T + 73iT^{2} \) |
| 79 | \( 1 + 16.3iT - 79T^{2} \) |
| 83 | \( 1 + (-0.171 + 0.171i)T - 83iT^{2} \) |
| 89 | \( 1 - 13.2T + 89T^{2} \) |
| 97 | \( 1 + (-12.6 + 12.6i)T - 97iT^{2} \) |
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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
−10.84089473689451321988500197772, −10.08926521483540480286638581159, −9.107625280700442414962519403250, −7.85192088818097206804196437284, −7.31570269424455716840439194606, −6.32218640323725288618269256081, −5.56245823870492659248820353523, −4.28172484281469495936181980860, −3.37021990011501530355805215957, −2.00717971249306768499807238979,
0.968447508978021015991705069809, 2.38525759269725103144732381289, 3.70991439583844926618068227685, 4.81943201014296890080476920453, 5.53283514156194697292255354003, 6.47127821317028424118608932521, 7.931588271283740476686399231496, 8.646950847089125534336555153727, 9.576030806473024286969734119144, 10.34516039817231120650618574808