L(s) = 1 | + 2-s + (−0.230 + 1.71i)3-s + 4-s + (0.286 + 2.21i)5-s + (−0.230 + 1.71i)6-s + 4.09i·7-s + 8-s + (−2.89 − 0.791i)9-s + (0.286 + 2.21i)10-s − 2.91·11-s + (−0.230 + 1.71i)12-s + 2.84·13-s + 4.09i·14-s + (−3.87 − 0.0193i)15-s + 16-s − 3.40i·17-s + ⋯ |
L(s) = 1 | + 0.707·2-s + (−0.133 + 0.991i)3-s + 0.5·4-s + (0.128 + 0.991i)5-s + (−0.0941 + 0.700i)6-s + 1.54i·7-s + 0.353·8-s + (−0.964 − 0.263i)9-s + (0.0906 + 0.701i)10-s − 0.879·11-s + (−0.0665 + 0.495i)12-s + 0.790·13-s + 1.09i·14-s + (−0.999 − 0.00499i)15-s + 0.250·16-s − 0.826i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.811 - 0.584i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.811 - 0.584i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.643722 + 1.99424i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.643722 + 1.99424i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 + (0.230 - 1.71i)T \) |
| 5 | \( 1 + (-0.286 - 2.21i)T \) |
| 31 | \( 1 + (-4.53 - 3.23i)T \) |
good | 7 | \( 1 - 4.09iT - 7T^{2} \) |
| 11 | \( 1 + 2.91T + 11T^{2} \) |
| 13 | \( 1 - 2.84T + 13T^{2} \) |
| 17 | \( 1 + 3.40iT - 17T^{2} \) |
| 19 | \( 1 - 2.22T + 19T^{2} \) |
| 23 | \( 1 + 3.74iT - 23T^{2} \) |
| 29 | \( 1 + 5.67T + 29T^{2} \) |
| 37 | \( 1 + 1.51T + 37T^{2} \) |
| 41 | \( 1 + 0.438iT - 41T^{2} \) |
| 43 | \( 1 - 8.29T + 43T^{2} \) |
| 47 | \( 1 - 7.81T + 47T^{2} \) |
| 53 | \( 1 - 10.9iT - 53T^{2} \) |
| 59 | \( 1 + 2.26iT - 59T^{2} \) |
| 61 | \( 1 - 3.00iT - 61T^{2} \) |
| 67 | \( 1 + 9.68iT - 67T^{2} \) |
| 71 | \( 1 - 11.2iT - 71T^{2} \) |
| 73 | \( 1 - 2.62T + 73T^{2} \) |
| 79 | \( 1 - 15.1iT - 79T^{2} \) |
| 83 | \( 1 - 8.73iT - 83T^{2} \) |
| 89 | \( 1 + 13.2T + 89T^{2} \) |
| 97 | \( 1 + 3.32iT - 97T^{2} \) |
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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
−10.62175036120847054629599218023, −9.649061710458201675110091575430, −8.891221624593525364478390998932, −7.899236145657182288455704059122, −6.69999234829177996076096656799, −5.70668965885185020544725304465, −5.40060555878944325665025143957, −4.14997718169561168306396799974, −2.95738817640733844736236696686, −2.51999786951729351968914939806,
0.795683973866890797191805812931, 1.86524869762202461954500031862, 3.44787896565018935405844398297, 4.37732262345242508682525198409, 5.47386605093419744824543027840, 6.12213756711154025854863884402, 7.30438786344317026540831026959, 7.76162046595848795004063770101, 8.628929615353904291378292228386, 9.883963772592810805544167160017