| L(s) = 1 | − 1.30·2-s + (−0.259 + 1.71i)3-s − 0.296·4-s − i·5-s + (0.338 − 2.23i)6-s − i·7-s + 2.99·8-s + (−2.86 − 0.888i)9-s + 1.30i·10-s + (3.24 − 0.692i)11-s + (0.0769 − 0.507i)12-s + 1.36i·13-s + 1.30i·14-s + (1.71 + 0.259i)15-s − 3.31·16-s − 2.94·17-s + ⋯ |
| L(s) = 1 | − 0.922·2-s + (−0.149 + 0.988i)3-s − 0.148·4-s − 0.447i·5-s + (0.138 − 0.912i)6-s − 0.377i·7-s + 1.05·8-s + (−0.955 − 0.296i)9-s + 0.412i·10-s + (0.977 − 0.208i)11-s + (0.0222 − 0.146i)12-s + 0.379i·13-s + 0.348i·14-s + (0.442 + 0.0669i)15-s − 0.829·16-s − 0.714·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0601 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0601 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4025262700\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4025262700\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.259 - 1.71i)T \) |
| 5 | \( 1 + iT \) |
| 7 | \( 1 + iT \) |
| 11 | \( 1 + (-3.24 + 0.692i)T \) |
| good | 2 | \( 1 + 1.30T + 2T^{2} \) |
| 13 | \( 1 - 1.36iT - 13T^{2} \) |
| 17 | \( 1 + 2.94T + 17T^{2} \) |
| 19 | \( 1 - 4.15iT - 19T^{2} \) |
| 23 | \( 1 + 5.83iT - 23T^{2} \) |
| 29 | \( 1 + 2.33T + 29T^{2} \) |
| 31 | \( 1 + 7.48T + 31T^{2} \) |
| 37 | \( 1 + 7.95T + 37T^{2} \) |
| 41 | \( 1 - 6.55T + 41T^{2} \) |
| 43 | \( 1 + 4.00iT - 43T^{2} \) |
| 47 | \( 1 + 4.08iT - 47T^{2} \) |
| 53 | \( 1 + 4.31iT - 53T^{2} \) |
| 59 | \( 1 - 2.85iT - 59T^{2} \) |
| 61 | \( 1 + 0.425iT - 61T^{2} \) |
| 67 | \( 1 - 1.24T + 67T^{2} \) |
| 71 | \( 1 + 3.21iT - 71T^{2} \) |
| 73 | \( 1 + 7.14iT - 73T^{2} \) |
| 79 | \( 1 + 11.0iT - 79T^{2} \) |
| 83 | \( 1 - 1.95T + 83T^{2} \) |
| 89 | \( 1 + 9.13iT - 89T^{2} \) |
| 97 | \( 1 + 0.814T + 97T^{2} \) |
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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
−9.422051162470477307131673643125, −8.937240459826474451232262980117, −8.395438790685305662296786058743, −7.29255821549360966092535748973, −6.26660475868534234529446734570, −5.16938897829869037331977829547, −4.26469045675585402885231001858, −3.69577693916927842557387617292, −1.80200149721506889958580690326, −0.27266438010997162759612838121,
1.26177469602260514180173987571, 2.31135177541356461546669329498, 3.69618416328259780449607599562, 5.02055859638626517790117652927, 6.00715428924777997304567721728, 7.03544811298201194489003943919, 7.43174977867457380804908954449, 8.448570219251506232752698049173, 9.111353163841651594718997255187, 9.693841142349022709969313928475
