L(s) = 1 | + 5-s − 2.82·7-s + 2.41·11-s + 1.82·13-s + 4.82·17-s − 6·19-s − 7.65·23-s − 4·25-s − 29-s + 4.07·31-s − 2.82·35-s − 4·37-s − 12.4·41-s − 6.41·43-s + 5.24·47-s + 1.00·49-s + 7.48·53-s + 2.41·55-s + 7.65·59-s + 0.828·61-s + 1.82·65-s + 5.65·67-s − 3.17·71-s + 4·73-s − 6.82·77-s − 0.414·79-s − 3.65·83-s + ⋯ |
L(s) = 1 | + 0.447·5-s − 1.06·7-s + 0.727·11-s + 0.507·13-s + 1.17·17-s − 1.37·19-s − 1.59·23-s − 0.800·25-s − 0.185·29-s + 0.731·31-s − 0.478·35-s − 0.657·37-s − 1.94·41-s − 0.978·43-s + 0.764·47-s + 0.142·49-s + 1.02·53-s + 0.325·55-s + 0.996·59-s + 0.106·61-s + 0.226·65-s + 0.691·67-s − 0.376·71-s + 0.468·73-s − 0.778·77-s − 0.0466·79-s − 0.401·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4176 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 29 | \( 1 + T \) |
good | 5 | \( 1 - T + 5T^{2} \) |
| 7 | \( 1 + 2.82T + 7T^{2} \) |
| 11 | \( 1 - 2.41T + 11T^{2} \) |
| 13 | \( 1 - 1.82T + 13T^{2} \) |
| 17 | \( 1 - 4.82T + 17T^{2} \) |
| 19 | \( 1 + 6T + 19T^{2} \) |
| 23 | \( 1 + 7.65T + 23T^{2} \) |
| 31 | \( 1 - 4.07T + 31T^{2} \) |
| 37 | \( 1 + 4T + 37T^{2} \) |
| 41 | \( 1 + 12.4T + 41T^{2} \) |
| 43 | \( 1 + 6.41T + 43T^{2} \) |
| 47 | \( 1 - 5.24T + 47T^{2} \) |
| 53 | \( 1 - 7.48T + 53T^{2} \) |
| 59 | \( 1 - 7.65T + 59T^{2} \) |
| 61 | \( 1 - 0.828T + 61T^{2} \) |
| 67 | \( 1 - 5.65T + 67T^{2} \) |
| 71 | \( 1 + 3.17T + 71T^{2} \) |
| 73 | \( 1 - 4T + 73T^{2} \) |
| 79 | \( 1 + 0.414T + 79T^{2} \) |
| 83 | \( 1 + 3.65T + 83T^{2} \) |
| 89 | \( 1 + 4.48T + 89T^{2} \) |
| 97 | \( 1 + 12.4T + 97T^{2} \) |
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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
−8.263863115154781886595602737943, −7.16519126413953215676834891696, −6.40535080919532336816534365480, −6.05223212462335313975471941026, −5.19836506744530229782616618662, −3.92678273195449574586909217301, −3.61129498391281736953451558606, −2.41290571547548827756643098894, −1.47601853651291160609701969231, 0,
1.47601853651291160609701969231, 2.41290571547548827756643098894, 3.61129498391281736953451558606, 3.92678273195449574586909217301, 5.19836506744530229782616618662, 6.05223212462335313975471941026, 6.40535080919532336816534365480, 7.16519126413953215676834891696, 8.263863115154781886595602737943