L(s) = 1 | − i·2-s + (0.707 + 0.707i)3-s − 4-s + (2.23 − 0.0687i)5-s + (0.707 − 0.707i)6-s + (2.13 + 2.13i)7-s + i·8-s + 1.00i·9-s + (−0.0687 − 2.23i)10-s − 2.07i·11-s + (−0.707 − 0.707i)12-s − 4.31i·13-s + (2.13 − 2.13i)14-s + (1.62 + 1.53i)15-s + 16-s + 2.62·17-s + ⋯ |
L(s) = 1 | − 0.707i·2-s + (0.408 + 0.408i)3-s − 0.5·4-s + (0.999 − 0.0307i)5-s + (0.288 − 0.288i)6-s + (0.807 + 0.807i)7-s + 0.353i·8-s + 0.333i·9-s + (−0.0217 − 0.706i)10-s − 0.625i·11-s + (−0.204 − 0.204i)12-s − 1.19i·13-s + (0.570 − 0.570i)14-s + (0.420 + 0.395i)15-s + 0.250·16-s + 0.635·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.899 + 0.436i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.899 + 0.436i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.345753521\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.345753521\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + iT \) |
| 3 | \( 1 + (-0.707 - 0.707i)T \) |
| 5 | \( 1 + (-2.23 + 0.0687i)T \) |
| 37 | \( 1 + (-5.47 + 2.65i)T \) |
good | 7 | \( 1 + (-2.13 - 2.13i)T + 7iT^{2} \) |
| 11 | \( 1 + 2.07iT - 11T^{2} \) |
| 13 | \( 1 + 4.31iT - 13T^{2} \) |
| 17 | \( 1 - 2.62T + 17T^{2} \) |
| 19 | \( 1 + (1.48 + 1.48i)T + 19iT^{2} \) |
| 23 | \( 1 - 3.49iT - 23T^{2} \) |
| 29 | \( 1 + (0.364 - 0.364i)T - 29iT^{2} \) |
| 31 | \( 1 + (-0.391 - 0.391i)T + 31iT^{2} \) |
| 41 | \( 1 - 3.80iT - 41T^{2} \) |
| 43 | \( 1 - 6.28iT - 43T^{2} \) |
| 47 | \( 1 + (-3.37 - 3.37i)T + 47iT^{2} \) |
| 53 | \( 1 + (1.87 - 1.87i)T - 53iT^{2} \) |
| 59 | \( 1 + (-0.917 - 0.917i)T + 59iT^{2} \) |
| 61 | \( 1 + (-5.87 - 5.87i)T + 61iT^{2} \) |
| 67 | \( 1 + (-3.05 + 3.05i)T - 67iT^{2} \) |
| 71 | \( 1 + 9.16T + 71T^{2} \) |
| 73 | \( 1 + (8.50 + 8.50i)T + 73iT^{2} \) |
| 79 | \( 1 + (9.01 + 9.01i)T + 79iT^{2} \) |
| 83 | \( 1 + (0.381 - 0.381i)T - 83iT^{2} \) |
| 89 | \( 1 + (3.68 - 3.68i)T - 89iT^{2} \) |
| 97 | \( 1 - 14.1T + 97T^{2} \) |
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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
−9.807825866142389626030704792904, −9.079817101485585586597575452509, −8.406700911455848296673956221631, −7.63557740263252328464535226997, −5.98338664058315287706714948250, −5.46877521799349786811468238940, −4.58838500834066038028764694588, −3.20461228508192364010496773535, −2.50753514225122387241331172767, −1.30908265236184960212443439597,
1.31162764050064284152926803663, 2.32228094662657698967985330077, 3.96207650624992181765507598445, 4.75787489261210000738541096335, 5.79617560881625609008432474319, 6.74851233560378804966902129091, 7.26682580005481500279188033592, 8.205875995907066939493375820884, 8.944656487541240314913707504475, 9.811065580226638245975177306745