L(s) = 1 | + 0.101·5-s + 0.950·7-s − 4.04·11-s + 4.96·13-s − 6.22·17-s + 1.02·19-s − 3.47·23-s − 4.98·25-s + 3.98·29-s − 4.26·31-s + 0.0963·35-s − 8.39·37-s − 2.80·41-s + 6.73·43-s + 7.82·47-s − 6.09·49-s − 12.9·53-s − 0.410·55-s + 14.2·59-s − 2.70·61-s + 0.503·65-s − 3.31·67-s − 14.5·71-s + 11.6·73-s − 3.84·77-s − 1.28·79-s − 5.25·83-s + ⋯ |
L(s) = 1 | + 0.0453·5-s + 0.359·7-s − 1.21·11-s + 1.37·13-s − 1.51·17-s + 0.236·19-s − 0.723·23-s − 0.997·25-s + 0.739·29-s − 0.766·31-s + 0.0162·35-s − 1.37·37-s − 0.438·41-s + 1.02·43-s + 1.14·47-s − 0.870·49-s − 1.78·53-s − 0.0553·55-s + 1.85·59-s − 0.346·61-s + 0.0624·65-s − 0.404·67-s − 1.72·71-s + 1.36·73-s − 0.438·77-s − 0.144·79-s − 0.576·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2916 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2916 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
good | 5 | \( 1 - 0.101T + 5T^{2} \) |
| 7 | \( 1 - 0.950T + 7T^{2} \) |
| 11 | \( 1 + 4.04T + 11T^{2} \) |
| 13 | \( 1 - 4.96T + 13T^{2} \) |
| 17 | \( 1 + 6.22T + 17T^{2} \) |
| 19 | \( 1 - 1.02T + 19T^{2} \) |
| 23 | \( 1 + 3.47T + 23T^{2} \) |
| 29 | \( 1 - 3.98T + 29T^{2} \) |
| 31 | \( 1 + 4.26T + 31T^{2} \) |
| 37 | \( 1 + 8.39T + 37T^{2} \) |
| 41 | \( 1 + 2.80T + 41T^{2} \) |
| 43 | \( 1 - 6.73T + 43T^{2} \) |
| 47 | \( 1 - 7.82T + 47T^{2} \) |
| 53 | \( 1 + 12.9T + 53T^{2} \) |
| 59 | \( 1 - 14.2T + 59T^{2} \) |
| 61 | \( 1 + 2.70T + 61T^{2} \) |
| 67 | \( 1 + 3.31T + 67T^{2} \) |
| 71 | \( 1 + 14.5T + 71T^{2} \) |
| 73 | \( 1 - 11.6T + 73T^{2} \) |
| 79 | \( 1 + 1.28T + 79T^{2} \) |
| 83 | \( 1 + 5.25T + 83T^{2} \) |
| 89 | \( 1 + 9.61T + 89T^{2} \) |
| 97 | \( 1 - 3.94T + 97T^{2} \) |
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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
−8.391097245082778398060736740168, −7.75578054070661848344459711570, −6.86277523416009251888131329987, −6.03822913630806822984038536361, −5.35331029506819669254317357370, −4.43404643136256998948896358176, −3.63243364200989964849972786215, −2.52032920588522096960765407470, −1.60677439816200762682381624192, 0,
1.60677439816200762682381624192, 2.52032920588522096960765407470, 3.63243364200989964849972786215, 4.43404643136256998948896358176, 5.35331029506819669254317357370, 6.03822913630806822984038536361, 6.86277523416009251888131329987, 7.75578054070661848344459711570, 8.391097245082778398060736740168