L(s) = 1 | + (0.923 + 0.382i)2-s + (0.707 − 0.707i)3-s + (0.707 + 0.707i)4-s + (0.382 + 0.923i)5-s + (0.923 − 0.382i)6-s + (0.382 + 0.923i)8-s − 1.00i·9-s + i·10-s + (−1.30 + 1.30i)11-s + 12-s + (0.707 − 0.707i)13-s + (0.923 + 0.382i)15-s + i·16-s + (0.382 − 0.923i)18-s + (−0.382 + 0.923i)20-s + ⋯ |
L(s) = 1 | + (0.923 + 0.382i)2-s + (0.707 − 0.707i)3-s + (0.707 + 0.707i)4-s + (0.382 + 0.923i)5-s + (0.923 − 0.382i)6-s + (0.382 + 0.923i)8-s − 1.00i·9-s + i·10-s + (−1.30 + 1.30i)11-s + 12-s + (0.707 − 0.707i)13-s + (0.923 + 0.382i)15-s + i·16-s + (0.382 − 0.923i)18-s + (−0.382 + 0.923i)20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.852437314\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.852437314\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.923 - 0.382i)T \) |
| 3 | \( 1 + (-0.707 + 0.707i)T \) |
| 5 | \( 1 + (-0.382 - 0.923i)T \) |
| 13 | \( 1 + (-0.707 + 0.707i)T \) |
good | 7 | \( 1 - T^{2} \) |
| 11 | \( 1 + (1.30 - 1.30i)T - iT^{2} \) |
| 17 | \( 1 + T^{2} \) |
| 19 | \( 1 - iT^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 - iT^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 - iT^{2} \) |
| 41 | \( 1 + 1.84iT - T^{2} \) |
| 43 | \( 1 + (-1 - i)T + iT^{2} \) |
| 47 | \( 1 + 1.84iT - T^{2} \) |
| 53 | \( 1 - iT^{2} \) |
| 59 | \( 1 + (-0.541 + 0.541i)T - iT^{2} \) |
| 61 | \( 1 + (1.41 + 1.41i)T + iT^{2} \) |
| 67 | \( 1 + iT^{2} \) |
| 71 | \( 1 - 0.765iT - T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 + 1.41T + T^{2} \) |
| 83 | \( 1 + (1.30 - 1.30i)T - iT^{2} \) |
| 89 | \( 1 + 0.765iT - T^{2} \) |
| 97 | \( 1 + T^{2} \) |
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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
−8.650144636672480137475738833194, −7.917896287548175260931460501899, −7.28822462224987682303540014321, −6.89588708907067993419784571900, −5.92126543100805348496007176935, −5.35727520844990187932934297848, −4.16628388343785862194499469418, −3.29126205557688277090930387446, −2.54345325332598135819449547759, −1.91225980549203786849195100201,
1.32000723934031762618414381869, 2.50436018317762594777189910020, 3.17178612559457716023693523149, 4.20187020976729418581568838255, 4.70042772871301153145054585203, 5.70587684928714186039453868907, 5.96569801102224019947283934441, 7.37399091404768200885423983699, 8.197022646147475918810791368565, 8.851704626305695778353564946381