L(s) = 1 | − 1.79i·3-s + 1.19i·5-s − 1.20i·7-s − 0.216·9-s − 0.564i·11-s − 6.78·13-s + 2.15·15-s + (−2.99 + 2.83i)17-s − 5.87·19-s − 2.16·21-s − 2.74i·23-s + 3.56·25-s − 4.99i·27-s + 8.94i·29-s − 4.79i·31-s + ⋯ |
L(s) = 1 | − 1.03i·3-s + 0.536i·5-s − 0.455i·7-s − 0.0720·9-s − 0.170i·11-s − 1.88·13-s + 0.555·15-s + (−0.727 + 0.686i)17-s − 1.34·19-s − 0.471·21-s − 0.571i·23-s + 0.712·25-s − 0.960i·27-s + 1.66i·29-s − 0.862i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.727 - 0.686i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.727 - 0.686i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.047937593\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.047937593\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 17 | \( 1 + (2.99 - 2.83i)T \) |
| 59 | \( 1 - T \) |
good | 3 | \( 1 + 1.79iT - 3T^{2} \) |
| 5 | \( 1 - 1.19iT - 5T^{2} \) |
| 7 | \( 1 + 1.20iT - 7T^{2} \) |
| 11 | \( 1 + 0.564iT - 11T^{2} \) |
| 13 | \( 1 + 6.78T + 13T^{2} \) |
| 19 | \( 1 + 5.87T + 19T^{2} \) |
| 23 | \( 1 + 2.74iT - 23T^{2} \) |
| 29 | \( 1 - 8.94iT - 29T^{2} \) |
| 31 | \( 1 + 4.79iT - 31T^{2} \) |
| 37 | \( 1 - 5.32iT - 37T^{2} \) |
| 41 | \( 1 - 6.25iT - 41T^{2} \) |
| 43 | \( 1 - 8.95T + 43T^{2} \) |
| 47 | \( 1 - 9.79T + 47T^{2} \) |
| 53 | \( 1 - 11.8T + 53T^{2} \) |
| 61 | \( 1 + 0.445iT - 61T^{2} \) |
| 67 | \( 1 + 13.0T + 67T^{2} \) |
| 71 | \( 1 + 9.22iT - 71T^{2} \) |
| 73 | \( 1 - 13.5iT - 73T^{2} \) |
| 79 | \( 1 - 15.6iT - 79T^{2} \) |
| 83 | \( 1 + 6.77T + 83T^{2} \) |
| 89 | \( 1 + 2.83T + 89T^{2} \) |
| 97 | \( 1 - 12.9iT - 97T^{2} \) |
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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
−8.407236785260992410922741439337, −7.57879587794586695496190094685, −7.00772730958273234538274635703, −6.66456273498949762735270734094, −5.77490518868662415979442895987, −4.65130487047085378301039617949, −4.07047837292203533066870655366, −2.65582775009713280413160698226, −2.25087304286201605123492366689, −0.988239257726855629426764499448,
0.33469224336114838642052407938, 2.10632246732894112635861748247, 2.73747982384821717327624266672, 4.16526453440980192392863159380, 4.43543206928410080029752330566, 5.21888865503205583664634647668, 5.88603258856100440656196314672, 7.13598329850461831076003759355, 7.45345139361523207207814822059, 8.850532582899476912970414685280