L(s) = 1 | + 5-s + 2·7-s + 11-s − 6.82·13-s − 1.17·17-s + 2.82·23-s + 25-s − 7.65·29-s + 2·35-s + 3.65·37-s − 6·41-s + 6·43-s − 2.82·47-s − 3·49-s − 0.343·53-s + 55-s − 9.65·59-s + 13.3·61-s − 6.82·65-s + 4.48·67-s − 11.3·71-s − 6.82·73-s + 2·77-s − 4·79-s − 6·83-s − 1.17·85-s − 9.31·89-s + ⋯ |
L(s) = 1 | + 0.447·5-s + 0.755·7-s + 0.301·11-s − 1.89·13-s − 0.284·17-s + 0.589·23-s + 0.200·25-s − 1.42·29-s + 0.338·35-s + 0.601·37-s − 0.937·41-s + 0.914·43-s − 0.412·47-s − 0.428·49-s − 0.0471·53-s + 0.134·55-s − 1.25·59-s + 1.70·61-s − 0.846·65-s + 0.547·67-s − 1.34·71-s − 0.799·73-s + 0.227·77-s − 0.450·79-s − 0.658·83-s − 0.127·85-s − 0.987·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 11 | \( 1 - T \) |
good | 7 | \( 1 - 2T + 7T^{2} \) |
| 13 | \( 1 + 6.82T + 13T^{2} \) |
| 17 | \( 1 + 1.17T + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 - 2.82T + 23T^{2} \) |
| 29 | \( 1 + 7.65T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 3.65T + 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 - 6T + 43T^{2} \) |
| 47 | \( 1 + 2.82T + 47T^{2} \) |
| 53 | \( 1 + 0.343T + 53T^{2} \) |
| 59 | \( 1 + 9.65T + 59T^{2} \) |
| 61 | \( 1 - 13.3T + 61T^{2} \) |
| 67 | \( 1 - 4.48T + 67T^{2} \) |
| 71 | \( 1 + 11.3T + 71T^{2} \) |
| 73 | \( 1 + 6.82T + 73T^{2} \) |
| 79 | \( 1 + 4T + 79T^{2} \) |
| 83 | \( 1 + 6T + 83T^{2} \) |
| 89 | \( 1 + 9.31T + 89T^{2} \) |
| 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
−7.39291328395012450288020043006, −6.99783575495615306952749849876, −6.05413838640723643893804521290, −5.27776512102457242011056553286, −4.79834693817511945426182034769, −4.05014062082837923559011409610, −2.93385558158188956855720097389, −2.20958152387170289383865167206, −1.41221950009749839239270445780, 0,
1.41221950009749839239270445780, 2.20958152387170289383865167206, 2.93385558158188956855720097389, 4.05014062082837923559011409610, 4.79834693817511945426182034769, 5.27776512102457242011056553286, 6.05413838640723643893804521290, 6.99783575495615306952749849876, 7.39291328395012450288020043006