L(s) = 1 | + (1.15 − 0.822i)2-s + (−1.99 − 1.99i)3-s + (0.647 − 1.89i)4-s + (2.01 − 2.01i)5-s + (−3.94 − 0.656i)6-s + 0.456i·7-s + (−0.810 − 2.70i)8-s + 4.98i·9-s + (0.662 − 3.97i)10-s + (−0.0937 + 0.0937i)11-s + (−5.07 + 2.48i)12-s + (3.81 + 3.81i)13-s + (0.375 + 0.525i)14-s − 8.06·15-s + (−3.16 − 2.45i)16-s − 5.00·17-s + ⋯ |
L(s) = 1 | + (0.813 − 0.581i)2-s + (−1.15 − 1.15i)3-s + (0.323 − 0.946i)4-s + (0.901 − 0.901i)5-s + (−1.60 − 0.267i)6-s + 0.172i·7-s + (−0.286 − 0.958i)8-s + 1.66i·9-s + (0.209 − 1.25i)10-s + (−0.0282 + 0.0282i)11-s + (−1.46 + 0.717i)12-s + (1.05 + 1.05i)13-s + (0.100 + 0.140i)14-s − 2.08·15-s + (−0.790 − 0.612i)16-s − 1.21·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.868 + 0.495i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.868 + 0.495i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.405491 - 1.52946i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.405491 - 1.52946i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.15 + 0.822i)T \) |
| 19 | \( 1 + (-0.707 - 0.707i)T \) |
good | 3 | \( 1 + (1.99 + 1.99i)T + 3iT^{2} \) |
| 5 | \( 1 + (-2.01 + 2.01i)T - 5iT^{2} \) |
| 7 | \( 1 - 0.456iT - 7T^{2} \) |
| 11 | \( 1 + (0.0937 - 0.0937i)T - 11iT^{2} \) |
| 13 | \( 1 + (-3.81 - 3.81i)T + 13iT^{2} \) |
| 17 | \( 1 + 5.00T + 17T^{2} \) |
| 23 | \( 1 + 3.03iT - 23T^{2} \) |
| 29 | \( 1 + (-4.15 - 4.15i)T + 29iT^{2} \) |
| 31 | \( 1 - 3.17T + 31T^{2} \) |
| 37 | \( 1 + (-8.33 + 8.33i)T - 37iT^{2} \) |
| 41 | \( 1 + 0.444iT - 41T^{2} \) |
| 43 | \( 1 + (3.04 - 3.04i)T - 43iT^{2} \) |
| 47 | \( 1 + 9.38T + 47T^{2} \) |
| 53 | \( 1 + (0.997 - 0.997i)T - 53iT^{2} \) |
| 59 | \( 1 + (-6.43 + 6.43i)T - 59iT^{2} \) |
| 61 | \( 1 + (7.30 + 7.30i)T + 61iT^{2} \) |
| 67 | \( 1 + (-8.26 - 8.26i)T + 67iT^{2} \) |
| 71 | \( 1 - 12.3iT - 71T^{2} \) |
| 73 | \( 1 - 4.07iT - 73T^{2} \) |
| 79 | \( 1 - 9.04T + 79T^{2} \) |
| 83 | \( 1 + (6.04 + 6.04i)T + 83iT^{2} \) |
| 89 | \( 1 + 9.07iT - 89T^{2} \) |
| 97 | \( 1 + 17.0T + 97T^{2} \) |
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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
−11.46172904871914467079920395747, −10.93095917849029626257402005839, −9.624150872723930185503969730342, −8.587964432542464310355206942037, −6.79965000141514407659399878840, −6.25282567792942246206264002942, −5.38382682190588555231973737017, −4.40250127932279973592789540904, −2.15566668041700517628672737426, −1.14373228230131347010467088424,
2.92148816994210938647449851224, 4.16629183996425943635897077861, 5.22216164622152665134131703168, 6.13814265610775931664717012765, 6.59645439919082913751782693513, 8.186773577036853678978473240619, 9.582432085534346152523895481581, 10.52162580969463540798352250481, 11.09172639557032617704156208252, 11.91177009285324990370343248393