L(s) = 1 | + 4·5-s − 2·7-s − 4·11-s − 5·13-s + 2·17-s − 6·19-s + 23-s + 11·25-s − 29-s + 9·31-s − 8·35-s − 4·37-s − 3·41-s − 8·43-s − 5·47-s − 3·49-s − 6·53-s − 16·55-s − 4·59-s − 10·61-s − 20·65-s + 4·67-s − 5·71-s − 15·73-s + 8·77-s + 6·79-s + 6·83-s + ⋯ |
L(s) = 1 | + 1.78·5-s − 0.755·7-s − 1.20·11-s − 1.38·13-s + 0.485·17-s − 1.37·19-s + 0.208·23-s + 11/5·25-s − 0.185·29-s + 1.61·31-s − 1.35·35-s − 0.657·37-s − 0.468·41-s − 1.21·43-s − 0.729·47-s − 3/7·49-s − 0.824·53-s − 2.15·55-s − 0.520·59-s − 1.28·61-s − 2.48·65-s + 0.488·67-s − 0.593·71-s − 1.75·73-s + 0.911·77-s + 0.675·79-s + 0.658·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3312 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 23 | \( 1 - T \) |
good | 5 | \( 1 - 4 T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + T + p T^{2} \) |
| 31 | \( 1 - 9 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 5 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 5 T + p T^{2} \) |
| 73 | \( 1 + 15 T + p T^{2} \) |
| 79 | \( 1 - 6 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 8 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−8.341766886235417246672414496569, −7.45524541986173157315507937524, −6.48575378238043427480742534693, −6.16811463932681713396370484318, −5.12110735176440726538257994756, −4.78073918536470462412494647478, −3.13411293269357723675344831334, −2.53958453578750887926295616856, −1.71700630586083558485724919296, 0,
1.71700630586083558485724919296, 2.53958453578750887926295616856, 3.13411293269357723675344831334, 4.78073918536470462412494647478, 5.12110735176440726538257994756, 6.16811463932681713396370484318, 6.48575378238043427480742534693, 7.45524541986173157315507937524, 8.341766886235417246672414496569