L(s) = 1 | − 2.41·2-s + 3.82·4-s − 2.82·5-s − 0.585·7-s − 4.41·8-s + 6.82·10-s + 2.82·11-s + 0.828·13-s + 1.41·14-s + 2.99·16-s + 2.24·17-s − 2.82·19-s − 10.8·20-s − 6.82·22-s + 4·23-s + 3.00·25-s − 1.99·26-s − 2.24·28-s − 8·29-s − 6.24·31-s + 1.58·32-s − 5.41·34-s + 1.65·35-s + 9.65·37-s + 6.82·38-s + 12.4·40-s − 4.58·41-s + ⋯ |
L(s) = 1 | − 1.70·2-s + 1.91·4-s − 1.26·5-s − 0.221·7-s − 1.56·8-s + 2.15·10-s + 0.852·11-s + 0.229·13-s + 0.377·14-s + 0.749·16-s + 0.543·17-s − 0.648·19-s − 2.42·20-s − 1.45·22-s + 0.834·23-s + 0.600·25-s − 0.392·26-s − 0.423·28-s − 1.48·29-s − 1.12·31-s + 0.280·32-s − 0.928·34-s + 0.280·35-s + 1.58·37-s + 1.10·38-s + 1.97·40-s − 0.716·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 639 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 639 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 71 | \( 1 + T \) |
good | 2 | \( 1 + 2.41T + 2T^{2} \) |
| 5 | \( 1 + 2.82T + 5T^{2} \) |
| 7 | \( 1 + 0.585T + 7T^{2} \) |
| 11 | \( 1 - 2.82T + 11T^{2} \) |
| 13 | \( 1 - 0.828T + 13T^{2} \) |
| 17 | \( 1 - 2.24T + 17T^{2} \) |
| 19 | \( 1 + 2.82T + 19T^{2} \) |
| 23 | \( 1 - 4T + 23T^{2} \) |
| 29 | \( 1 + 8T + 29T^{2} \) |
| 31 | \( 1 + 6.24T + 31T^{2} \) |
| 37 | \( 1 - 9.65T + 37T^{2} \) |
| 41 | \( 1 + 4.58T + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 + 4T + 47T^{2} \) |
| 53 | \( 1 - 4.58T + 53T^{2} \) |
| 59 | \( 1 + 10.8T + 59T^{2} \) |
| 61 | \( 1 + 3.17T + 61T^{2} \) |
| 67 | \( 1 + 7.89T + 67T^{2} \) |
| 73 | \( 1 + 13.6T + 73T^{2} \) |
| 79 | \( 1 + 2.82T + 79T^{2} \) |
| 83 | \( 1 + 16.9T + 83T^{2} \) |
| 89 | \( 1 + 5.17T + 89T^{2} \) |
| 97 | \( 1 + 7.65T + 97T^{2} \) |
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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
−9.946845748946493581565713732419, −9.163580879009476396766551636077, −8.501083895772061777499313140995, −7.62232689699973169375747456005, −7.09283253132656684255341238247, −6.01817617618368834129059041709, −4.30016587650759787263206104224, −3.19397670804013803843642142220, −1.50799979049444614210788979200, 0,
1.50799979049444614210788979200, 3.19397670804013803843642142220, 4.30016587650759787263206104224, 6.01817617618368834129059041709, 7.09283253132656684255341238247, 7.62232689699973169375747456005, 8.501083895772061777499313140995, 9.163580879009476396766551636077, 9.946845748946493581565713732419