L(s) = 1 | + (0.707 + 0.707i)3-s + (−0.707 + 0.707i)5-s + 11-s + (−0.707 − 0.707i)13-s − 1.00·15-s + (−0.707 + 0.707i)17-s + 1.41i·19-s + (1 + i)23-s − 1.00i·25-s + (0.707 − 0.707i)27-s + i·29-s + (0.707 + 0.707i)33-s + (−1 + i)37-s − 1.00i·39-s + (1 + i)43-s + ⋯ |
L(s) = 1 | + (0.707 + 0.707i)3-s + (−0.707 + 0.707i)5-s + 11-s + (−0.707 − 0.707i)13-s − 1.00·15-s + (−0.707 + 0.707i)17-s + 1.41i·19-s + (1 + i)23-s − 1.00i·25-s + (0.707 − 0.707i)27-s + i·29-s + (0.707 + 0.707i)33-s + (−1 + i)37-s − 1.00i·39-s + (1 + i)43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.229 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.229 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.360914172\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.360914172\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (0.707 - 0.707i)T \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + (-0.707 - 0.707i)T + iT^{2} \) |
| 11 | \( 1 - T + T^{2} \) |
| 13 | \( 1 + (0.707 + 0.707i)T + iT^{2} \) |
| 17 | \( 1 + (0.707 - 0.707i)T - iT^{2} \) |
| 19 | \( 1 - 1.41iT - T^{2} \) |
| 23 | \( 1 + (-1 - i)T + iT^{2} \) |
| 29 | \( 1 - iT - T^{2} \) |
| 31 | \( 1 + T^{2} \) |
| 37 | \( 1 + (1 - i)T - iT^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 + (-1 - i)T + iT^{2} \) |
| 47 | \( 1 + (0.707 - 0.707i)T - iT^{2} \) |
| 53 | \( 1 + iT^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 + T^{2} \) |
| 67 | \( 1 - iT^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 + iT^{2} \) |
| 79 | \( 1 + iT - T^{2} \) |
| 83 | \( 1 + iT^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + (-0.707 + 0.707i)T - iT^{2} \) |
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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
−8.843522838957131731790203604113, −8.198051813973486014735902122482, −7.48573882039519917450708814385, −6.71778996885544165869776609327, −6.01650509900396672636972589621, −4.87564060655328401226414367512, −4.07280437800404577527269942575, −3.42387560408457610987567007966, −2.91633559672192806992462068900, −1.51680575089813441317902430847,
0.72641337591353147999404367569, 2.01329573705601649636190510124, 2.71179751438656202336865655268, 3.87959741459263097284528336055, 4.62473604242831414998589507190, 5.21302924320673264883171717473, 6.67759146117564108056903845575, 7.00315067327827743935720095742, 7.64427606345789594299610872707, 8.567944625962545499314946539046