L(s) = 1 | + 3.16·3-s − 2.82·5-s + 7.00·9-s + 4.47·11-s − 8.94·15-s − 4.24·17-s + 3.16·19-s + 8.94·23-s + 3.00·25-s + 12.6·27-s − 6·29-s + 6.32·31-s + 14.1·33-s + 2·37-s − 1.41·41-s − 4.47·43-s − 19.7·45-s − 6.32·47-s − 13.4·51-s + 6·53-s − 12.6·55-s + 10.0·57-s + 3.16·59-s + 8.48·61-s + 28.2·69-s + 8.94·71-s + 7.07·73-s + ⋯ |
L(s) = 1 | + 1.82·3-s − 1.26·5-s + 2.33·9-s + 1.34·11-s − 2.30·15-s − 1.02·17-s + 0.725·19-s + 1.86·23-s + 0.600·25-s + 2.43·27-s − 1.11·29-s + 1.13·31-s + 2.46·33-s + 0.328·37-s − 0.220·41-s − 0.681·43-s − 2.95·45-s − 0.922·47-s − 1.87·51-s + 0.824·53-s − 1.70·55-s + 1.32·57-s + 0.411·59-s + 1.08·61-s + 3.40·69-s + 1.06·71-s + 0.827·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.931883303\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.931883303\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
good | 3 | \( 1 - 3.16T + 3T^{2} \) |
| 5 | \( 1 + 2.82T + 5T^{2} \) |
| 11 | \( 1 - 4.47T + 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 + 4.24T + 17T^{2} \) |
| 19 | \( 1 - 3.16T + 19T^{2} \) |
| 23 | \( 1 - 8.94T + 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 - 6.32T + 31T^{2} \) |
| 37 | \( 1 - 2T + 37T^{2} \) |
| 41 | \( 1 + 1.41T + 41T^{2} \) |
| 43 | \( 1 + 4.47T + 43T^{2} \) |
| 47 | \( 1 + 6.32T + 47T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 - 3.16T + 59T^{2} \) |
| 61 | \( 1 - 8.48T + 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 - 8.94T + 71T^{2} \) |
| 73 | \( 1 - 7.07T + 73T^{2} \) |
| 79 | \( 1 - 8.94T + 79T^{2} \) |
| 83 | \( 1 + 9.48T + 83T^{2} \) |
| 89 | \( 1 + 9.89T + 89T^{2} \) |
| 97 | \( 1 + 4.24T + 97T^{2} \) |
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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
−9.233661505903103261617081367516, −8.586696222577416729906747946593, −8.039711473485126745942232495043, −7.11957122394869153554876539063, −6.74389614706836314468046805975, −4.89860461005256719269545343341, −3.98354626339970453753901510572, −3.52772005346335463249579889641, −2.56336977921304448822556961718, −1.23629472945942244569028190055,
1.23629472945942244569028190055, 2.56336977921304448822556961718, 3.52772005346335463249579889641, 3.98354626339970453753901510572, 4.89860461005256719269545343341, 6.74389614706836314468046805975, 7.11957122394869153554876539063, 8.039711473485126745942232495043, 8.586696222577416729906747946593, 9.233661505903103261617081367516