| L(s) = 1 | + i·3-s − i·5-s + 3.74·7-s − 9-s − 0.951·11-s − 0.958·13-s + 15-s − 7.12i·17-s + 8.14·19-s + 3.74i·21-s + (−3.34 + 3.43i)23-s − 25-s − i·27-s + 3.97·29-s + 10.2i·31-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s − 0.447i·5-s + 1.41·7-s − 0.333·9-s − 0.286·11-s − 0.265·13-s + 0.258·15-s − 1.72i·17-s + 1.86·19-s + 0.818i·21-s + (−0.697 + 0.716i)23-s − 0.200·25-s − 0.192i·27-s + 0.738·29-s + 1.84i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.969 + 0.245i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.969 + 0.245i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.358723695\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.358723695\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 + iT \) |
| 23 | \( 1 + (3.34 - 3.43i)T \) |
| good | 7 | \( 1 - 3.74T + 7T^{2} \) |
| 11 | \( 1 + 0.951T + 11T^{2} \) |
| 13 | \( 1 + 0.958T + 13T^{2} \) |
| 17 | \( 1 + 7.12iT - 17T^{2} \) |
| 19 | \( 1 - 8.14T + 19T^{2} \) |
| 29 | \( 1 - 3.97T + 29T^{2} \) |
| 31 | \( 1 - 10.2iT - 31T^{2} \) |
| 37 | \( 1 + 7.97iT - 37T^{2} \) |
| 41 | \( 1 - 2.05T + 41T^{2} \) |
| 43 | \( 1 + 4.47T + 43T^{2} \) |
| 47 | \( 1 + 5.73iT - 47T^{2} \) |
| 53 | \( 1 - 5.84iT - 53T^{2} \) |
| 59 | \( 1 + 5.24iT - 59T^{2} \) |
| 61 | \( 1 - 1.55iT - 61T^{2} \) |
| 67 | \( 1 - 8.31T + 67T^{2} \) |
| 71 | \( 1 + 14.9iT - 71T^{2} \) |
| 73 | \( 1 - 2.66T + 73T^{2} \) |
| 79 | \( 1 - 15.4T + 79T^{2} \) |
| 83 | \( 1 + 3.86T + 83T^{2} \) |
| 89 | \( 1 + 3.08iT - 89T^{2} \) |
| 97 | \( 1 + 5.79iT - 97T^{2} \) |
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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
−7.998793099310741080103917985879, −7.62773090214865454829724247888, −6.86147774422874450798653875264, −5.56345278539607318491026166320, −5.04925537694049666841722059334, −4.81893347226895461386466135667, −3.68032686715389136160816756984, −2.85408023133631629342393593180, −1.78077496969332305794861452683, −0.74000394683989349447062340767,
1.00460070060097486025528646334, 1.88492726766312181592354497892, 2.65952866028122034282110288322, 3.72319740607902587565419372194, 4.57161827817457995970569218775, 5.36733595160442925228017348222, 6.05148863513173010823510099268, 6.78879140687126025392647409046, 7.70836500830553917081189655316, 8.032700858875141244590882644694
