L(s) = 1 | + (1.14 + 1.98i)3-s + (1.96 − 3.39i)5-s + (−0.412 − 2.61i)7-s + (−1.12 + 1.95i)9-s + (−1.75 − 3.04i)11-s − 3.61·13-s + 8.99·15-s + (3.34 + 5.79i)17-s + (−0.103 + 0.179i)19-s + (4.71 − 3.81i)21-s + (1.39 − 2.41i)23-s + (−5.19 − 8.99i)25-s + 1.71·27-s − 2.98·29-s + (−2.14 − 3.72i)31-s + ⋯ |
L(s) = 1 | + (0.661 + 1.14i)3-s + (0.877 − 1.51i)5-s + (−0.155 − 0.987i)7-s + (−0.375 + 0.650i)9-s + (−0.529 − 0.917i)11-s − 1.00·13-s + 2.32·15-s + (0.811 + 1.40i)17-s + (−0.0237 + 0.0411i)19-s + (1.02 − 0.831i)21-s + (0.290 − 0.503i)23-s + (−1.03 − 1.79i)25-s + 0.329·27-s − 0.554·29-s + (−0.385 − 0.668i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.676 + 0.736i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.676 + 0.736i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.124390905\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.124390905\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + (0.412 + 2.61i)T \) |
| 41 | \( 1 + T \) |
good | 3 | \( 1 + (-1.14 - 1.98i)T + (-1.5 + 2.59i)T^{2} \) |
| 5 | \( 1 + (-1.96 + 3.39i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (1.75 + 3.04i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + 3.61T + 13T^{2} \) |
| 17 | \( 1 + (-3.34 - 5.79i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (0.103 - 0.179i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.39 + 2.41i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 2.98T + 29T^{2} \) |
| 31 | \( 1 + (2.14 + 3.72i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-5.14 + 8.90i)T + (-18.5 - 32.0i)T^{2} \) |
| 43 | \( 1 - 9.05T + 43T^{2} \) |
| 47 | \( 1 + (-5.26 + 9.11i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-6.35 - 11.0i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (4.06 + 7.03i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (1.90 - 3.30i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-2.29 - 3.97i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 5.96T + 71T^{2} \) |
| 73 | \( 1 + (-7.03 - 12.1i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-6.17 + 10.6i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 5.55T + 83T^{2} \) |
| 89 | \( 1 + (1.16 - 2.01i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 8.84T + 97T^{2} \) |
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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
−9.592798536404643323174553796404, −9.063115771763597167155065562051, −8.303179727081859297282964426553, −7.50904176847298913314402697888, −5.95998019420261553889579886120, −5.33184202203623319586275024858, −4.33873717604317174077448412848, −3.76736781739050535025531133299, −2.38869217462589660625868104975, −0.850819167544585156853834300589,
1.80463087387018941950716158119, 2.67992848508283382107893024725, 2.94358408335689166811283006557, 5.00667812769764269348778982844, 5.86330338581860429457911060283, 6.88505450243839352250193228222, 7.28651727854691024152396733555, 7.958059444572824494364871131441, 9.356379454750020593128732065491, 9.667979739303150157015816585041