L(s) = 1 | + (−0.923 + 0.923i)3-s + (−2.48 − 0.915i)7-s + 1.29i·9-s + 1.21·11-s + (−0.996 + 0.996i)13-s + (−0.567 − 0.567i)17-s − 0.103·19-s + (3.13 − 1.44i)21-s + (−1.43 − 1.43i)23-s + (−3.96 − 3.96i)27-s + 4.97i·29-s − 4.35i·31-s + (−1.12 + 1.12i)33-s + (2.54 − 2.54i)37-s − 1.83i·39-s + ⋯ |
L(s) = 1 | + (−0.532 + 0.532i)3-s + (−0.938 − 0.345i)7-s + 0.431i·9-s + 0.366·11-s + (−0.276 + 0.276i)13-s + (−0.137 − 0.137i)17-s − 0.0237·19-s + (0.684 − 0.315i)21-s + (−0.298 − 0.298i)23-s + (−0.763 − 0.763i)27-s + 0.924i·29-s − 0.781i·31-s + (−0.195 + 0.195i)33-s + (0.417 − 0.417i)37-s − 0.294i·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.220 + 0.975i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.220 + 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4168272456\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4168272456\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (2.48 + 0.915i)T \) |
good | 3 | \( 1 + (0.923 - 0.923i)T - 3iT^{2} \) |
| 11 | \( 1 - 1.21T + 11T^{2} \) |
| 13 | \( 1 + (0.996 - 0.996i)T - 13iT^{2} \) |
| 17 | \( 1 + (0.567 + 0.567i)T + 17iT^{2} \) |
| 19 | \( 1 + 0.103T + 19T^{2} \) |
| 23 | \( 1 + (1.43 + 1.43i)T + 23iT^{2} \) |
| 29 | \( 1 - 4.97iT - 29T^{2} \) |
| 31 | \( 1 + 4.35iT - 31T^{2} \) |
| 37 | \( 1 + (-2.54 + 2.54i)T - 37iT^{2} \) |
| 41 | \( 1 + 7.06iT - 41T^{2} \) |
| 43 | \( 1 + (3.11 + 3.11i)T + 43iT^{2} \) |
| 47 | \( 1 + (7.23 + 7.23i)T + 47iT^{2} \) |
| 53 | \( 1 + (0.882 + 0.882i)T + 53iT^{2} \) |
| 59 | \( 1 + 8.28T + 59T^{2} \) |
| 61 | \( 1 + 10.0iT - 61T^{2} \) |
| 67 | \( 1 + (-7.07 + 7.07i)T - 67iT^{2} \) |
| 71 | \( 1 - 0.329T + 71T^{2} \) |
| 73 | \( 1 + (11.3 - 11.3i)T - 73iT^{2} \) |
| 79 | \( 1 + 13.6iT - 79T^{2} \) |
| 83 | \( 1 + (4.61 - 4.61i)T - 83iT^{2} \) |
| 89 | \( 1 - 13.1T + 89T^{2} \) |
| 97 | \( 1 + (-2.40 - 2.40i)T + 97iT^{2} \) |
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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
−9.527867041609732749456303237377, −8.674295610067667499148271409855, −7.60668807970463525083783753590, −6.80851495798383290195538466077, −6.01950249781205990733121868313, −5.10455722885609861412326199590, −4.24563097163799463555979708091, −3.36069156608663259702263297291, −2.04831897769660018929444416825, −0.18993483627494238534008572391,
1.26831193487252739378758187129, 2.73965641562164256098725989506, 3.67559329968801363044146545419, 4.84384784069474724469315357763, 6.03767544012496378393115817091, 6.31892955576346338708799773038, 7.21241554590611583123012227648, 8.114925567792465624278288151247, 9.147974125168837077346968727281, 9.700004206065555043673158124143