L(s) = 1 | + 5-s − 3.70·7-s + 5.70·11-s − 13-s − 5.70·17-s − 2·19-s − 0.298·23-s + 25-s + 3.40·29-s + 3.40·31-s − 3.70·35-s − 0.298·37-s − 4.29·41-s + 4·43-s − 11.4·47-s + 6.70·49-s − 4.29·53-s + 5.70·55-s − 10.8·59-s − 3.70·61-s − 65-s − 4·67-s + 16.5·71-s + 0.596·73-s − 21.1·77-s − 1.70·79-s − 4·83-s + ⋯ |
L(s) = 1 | + 0.447·5-s − 1.39·7-s + 1.71·11-s − 0.277·13-s − 1.38·17-s − 0.458·19-s − 0.0622·23-s + 0.200·25-s + 0.631·29-s + 0.611·31-s − 0.625·35-s − 0.0490·37-s − 0.671·41-s + 0.609·43-s − 1.66·47-s + 0.957·49-s − 0.590·53-s + 0.768·55-s − 1.40·59-s − 0.473·61-s − 0.124·65-s − 0.488·67-s + 1.95·71-s + 0.0698·73-s − 2.40·77-s − 0.191·79-s − 0.439·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 + T \) |
good | 7 | \( 1 + 3.70T + 7T^{2} \) |
| 11 | \( 1 - 5.70T + 11T^{2} \) |
| 17 | \( 1 + 5.70T + 17T^{2} \) |
| 19 | \( 1 + 2T + 19T^{2} \) |
| 23 | \( 1 + 0.298T + 23T^{2} \) |
| 29 | \( 1 - 3.40T + 29T^{2} \) |
| 31 | \( 1 - 3.40T + 31T^{2} \) |
| 37 | \( 1 + 0.298T + 37T^{2} \) |
| 41 | \( 1 + 4.29T + 41T^{2} \) |
| 43 | \( 1 - 4T + 43T^{2} \) |
| 47 | \( 1 + 11.4T + 47T^{2} \) |
| 53 | \( 1 + 4.29T + 53T^{2} \) |
| 59 | \( 1 + 10.8T + 59T^{2} \) |
| 61 | \( 1 + 3.70T + 61T^{2} \) |
| 67 | \( 1 + 4T + 67T^{2} \) |
| 71 | \( 1 - 16.5T + 71T^{2} \) |
| 73 | \( 1 - 0.596T + 73T^{2} \) |
| 79 | \( 1 + 1.70T + 79T^{2} \) |
| 83 | \( 1 + 4T + 83T^{2} \) |
| 89 | \( 1 - 11.1T + 89T^{2} \) |
| 97 | \( 1 + 1.10T + 97T^{2} \) |
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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
−8.012120298836959585396917209882, −6.74182842888369090230001590045, −6.61574962888952424884211303934, −6.09416029570506197015143689000, −4.88355066272775390497466452330, −4.13537722801390120476438690334, −3.34520180516445594332324943577, −2.46328844387722099136156912619, −1.40181083449293353788992308621, 0,
1.40181083449293353788992308621, 2.46328844387722099136156912619, 3.34520180516445594332324943577, 4.13537722801390120476438690334, 4.88355066272775390497466452330, 6.09416029570506197015143689000, 6.61574962888952424884211303934, 6.74182842888369090230001590045, 8.012120298836959585396917209882