| L(s) = 1 | + 3-s + 5-s − i·7-s − i·11-s + 15-s − i·21-s + 25-s − 27-s + (−1 − i)31-s − i·33-s − i·35-s + i·37-s + i·41-s + (1 + i)43-s + i·47-s + ⋯ |
| L(s) = 1 | + 3-s + 5-s − i·7-s − i·11-s + 15-s − i·21-s + 25-s − 27-s + (−1 − i)31-s − i·33-s − i·35-s + i·37-s + i·41-s + (1 + i)43-s + i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.763 + 0.646i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.763 + 0.646i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.972410737\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.972410737\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 37 | \( 1 - iT \) |
| good | 3 | \( 1 - T + T^{2} \) |
| 7 | \( 1 + iT - T^{2} \) |
| 11 | \( 1 + iT - T^{2} \) |
| 13 | \( 1 + iT^{2} \) |
| 17 | \( 1 + iT^{2} \) |
| 19 | \( 1 - iT^{2} \) |
| 23 | \( 1 + iT^{2} \) |
| 29 | \( 1 + iT^{2} \) |
| 31 | \( 1 + (1 + i)T + iT^{2} \) |
| 41 | \( 1 - iT - T^{2} \) |
| 43 | \( 1 + (-1 - i)T + iT^{2} \) |
| 47 | \( 1 - iT - T^{2} \) |
| 53 | \( 1 + iT - T^{2} \) |
| 59 | \( 1 - iT^{2} \) |
| 61 | \( 1 + (1 + i)T + iT^{2} \) |
| 67 | \( 1 + T^{2} \) |
| 71 | \( 1 - T + T^{2} \) |
| 73 | \( 1 - T + T^{2} \) |
| 79 | \( 1 + (1 - i)T - iT^{2} \) |
| 83 | \( 1 - iT - T^{2} \) |
| 89 | \( 1 + (1 + i)T + iT^{2} \) |
| 97 | \( 1 + (-1 - i)T + iT^{2} \) |
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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
−8.885091469160652560019143267477, −8.103330691581431595383593909047, −7.55514317719899712486690314678, −6.50641991675492152299541243899, −5.92413727805773491659101914419, −4.95897654898646017536991845606, −3.90586087607781531562269818988, −3.14461770674360719510883973381, −2.33499886408065518786963911309, −1.16579340363449664436353293054,
1.83618408207245179456766491988, 2.31855443981185852353337202802, 3.16811921201409471414787485269, 4.24333617522462865527958488157, 5.41707648081504846016104737054, 5.72944983149797268441627081549, 6.89223291455868412644307205074, 7.50542736051155391613598042313, 8.554746379815246561482071674591, 9.059475940381516885011002168660
