L(s) = 1 | − 2-s − 3-s + 4-s − 5-s + 6-s − 7-s − 8-s + 9-s + 10-s − 11-s − 12-s − 13-s + 14-s + 15-s + 16-s + 17-s − 18-s + 19-s − 20-s + 21-s + 22-s − 23-s + 24-s + 25-s + 26-s − 27-s − 28-s + ⋯ |
L(s) = 1 | − 2-s − 3-s + 4-s − 5-s + 6-s − 7-s − 8-s + 9-s + 10-s − 11-s − 12-s − 13-s + 14-s + 15-s + 16-s + 17-s − 18-s + 19-s − 20-s + 21-s + 22-s − 23-s + 24-s + 25-s + 26-s − 27-s − 28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2347 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2347 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2241703546\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2241703546\) |
\(L(1)\) |
\(\approx\) |
\(0.3242375672\) |
\(L(1)\) |
\(\approx\) |
\(0.3242375672\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2347 | \( 1 \) |
good | 2 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 - T \) |
| 31 | \( 1 + T \) |
| 37 | \( 1 + T \) |
| 41 | \( 1 + T \) |
| 43 | \( 1 - T \) |
| 47 | \( 1 + T \) |
| 53 | \( 1 + T \) |
| 59 | \( 1 - T \) |
| 61 | \( 1 - T \) |
| 67 | \( 1 + T \) |
| 71 | \( 1 - T \) |
| 73 | \( 1 - T \) |
| 79 | \( 1 + T \) |
| 83 | \( 1 - T \) |
| 89 | \( 1 - T \) |
| 97 | \( 1 - T \) |
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\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−19.26238750951747940034525118707, −18.53110978101683251773073702989, −18.23590322554603183135405817374, −17.12273018988534956309414500531, −16.56890187869767345439327146349, −16.028519140662761242372638581591, −15.53675479497193362699825749100, −14.77659877707709609220974026438, −13.40820783294972308459273554587, −12.30867939448506942221764431040, −12.22445208367164802434921492637, −11.37172866868655056121288873551, −10.54423426579633705314149606150, −9.92529349670977574585001617144, −9.43622168603709345085744298502, −8.11247817528232230190539988955, −7.50206425817934665161547728802, −7.08874974376451551440628365982, −6.00284316773688143096675347039, −5.43418950141067023410110458206, −4.30285950389929486487855396059, −3.272941065950403982466572107840, −2.50940337081810409979670948162, −1.10697331693776989043860215758, −0.269873707217955706046646639232,
0.269873707217955706046646639232, 1.10697331693776989043860215758, 2.50940337081810409979670948162, 3.272941065950403982466572107840, 4.30285950389929486487855396059, 5.43418950141067023410110458206, 6.00284316773688143096675347039, 7.08874974376451551440628365982, 7.50206425817934665161547728802, 8.11247817528232230190539988955, 9.43622168603709345085744298502, 9.92529349670977574585001617144, 10.54423426579633705314149606150, 11.37172866868655056121288873551, 12.22445208367164802434921492637, 12.30867939448506942221764431040, 13.40820783294972308459273554587, 14.77659877707709609220974026438, 15.53675479497193362699825749100, 16.028519140662761242372638581591, 16.56890187869767345439327146349, 17.12273018988534956309414500531, 18.23590322554603183135405817374, 18.53110978101683251773073702989, 19.26238750951747940034525118707