L(s) = 1 | + (−1 + i)2-s − i·4-s + i·5-s + i·9-s + (−1 − i)10-s + 11-s + (1 + i)13-s + 16-s + (−1 − i)18-s − i·19-s + 20-s + (−1 + i)22-s + (−1 − i)23-s − 25-s − 2·26-s + ⋯ |
L(s) = 1 | + (−1 + i)2-s − i·4-s + i·5-s + i·9-s + (−1 − i)10-s + 11-s + (1 + i)13-s + 16-s + (−1 − i)18-s − i·19-s + 20-s + (−1 + i)22-s + (−1 − i)23-s − 25-s − 2·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.850 - 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.850 - 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6145859214\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6145859214\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 - iT \) |
| 11 | \( 1 - T \) |
| 19 | \( 1 + iT \) |
good | 2 | \( 1 + (1 - i)T - iT^{2} \) |
| 3 | \( 1 - iT^{2} \) |
| 7 | \( 1 + iT^{2} \) |
| 13 | \( 1 + (-1 - i)T + iT^{2} \) |
| 17 | \( 1 + iT^{2} \) |
| 23 | \( 1 + (1 + i)T + iT^{2} \) |
| 29 | \( 1 - 2iT - T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + iT^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 - iT^{2} \) |
| 47 | \( 1 + (1 - i)T - iT^{2} \) |
| 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 | \( 1 - T^{2} \) |
| 83 | \( 1 - iT^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 + iT^{2} \) |
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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
−10.33475460401745601792993923074, −9.397071014794155632236721369133, −8.707675254976155888493077512234, −8.013869389305407121251535938426, −6.88366514467458759264822792460, −6.75263552876485218216078419294, −5.77642914467062816249419478091, −4.40768063736441103055922973967, −3.24554637044310628553852979076, −1.75206183071507496831304329491,
0.868387995510878913450839233923, 1.79926170213146696960084929396, 3.42725120732277916088107447536, 4.05035672074264593152838961880, 5.68826381795151903456043795518, 6.19626198096216368649407430673, 7.85204055585456626447240482814, 8.333479885990558255116293997436, 9.190234876684105935306729063950, 9.693456368266116561304895484405