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
−9.811065580226638245975177306745, −8.944656487541240314913707504475, −8.205875995907066939493375820884, −7.26682580005481500279188033592, −6.74851233560378804966902129091, −5.79617560881625609008432474319, −4.75787489261210000738541096335, −3.96207650624992181765507598445, −2.32228094662657698967985330077, −1.31162764050064284152926803663,
1.30908265236184960212443439597, 2.50753514225122387241331172767, 3.20461228508192364010496773535, 4.58838500834066038028764694588, 5.46877521799349786811468238940, 5.98338664058315287706714948250, 7.63557740263252328464535226997, 8.406700911455848296673956221631, 9.079817101485585586597575452509, 9.807825866142389626030704792904