L(s) = 1 | + 12·5-s − 2·7-s − 36·11-s + 13·13-s + 78·17-s − 74·19-s − 96·23-s + 19·25-s − 18·29-s + 214·31-s − 24·35-s − 286·37-s + 384·41-s − 524·43-s + 300·47-s − 339·49-s − 558·53-s − 432·55-s + 576·59-s + 74·61-s + 156·65-s − 38·67-s − 456·71-s − 682·73-s + 72·77-s − 704·79-s − 888·83-s + ⋯ |
L(s) = 1 | + 1.07·5-s − 0.107·7-s − 0.986·11-s + 0.277·13-s + 1.11·17-s − 0.893·19-s − 0.870·23-s + 0.151·25-s − 0.115·29-s + 1.23·31-s − 0.115·35-s − 1.27·37-s + 1.46·41-s − 1.85·43-s + 0.931·47-s − 0.988·49-s − 1.44·53-s − 1.05·55-s + 1.27·59-s + 0.155·61-s + 0.297·65-s − 0.0692·67-s − 0.762·71-s − 1.09·73-s + 0.106·77-s − 1.00·79-s − 1.17·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 13 | \( 1 - p T \) |
good | 5 | \( 1 - 12 T + p^{3} T^{2} \) |
| 7 | \( 1 + 2 T + p^{3} T^{2} \) |
| 11 | \( 1 + 36 T + p^{3} T^{2} \) |
| 17 | \( 1 - 78 T + p^{3} T^{2} \) |
| 19 | \( 1 + 74 T + p^{3} T^{2} \) |
| 23 | \( 1 + 96 T + p^{3} T^{2} \) |
| 29 | \( 1 + 18 T + p^{3} T^{2} \) |
| 31 | \( 1 - 214 T + p^{3} T^{2} \) |
| 37 | \( 1 + 286 T + p^{3} T^{2} \) |
| 41 | \( 1 - 384 T + p^{3} T^{2} \) |
| 43 | \( 1 + 524 T + p^{3} T^{2} \) |
| 47 | \( 1 - 300 T + p^{3} T^{2} \) |
| 53 | \( 1 + 558 T + p^{3} T^{2} \) |
| 59 | \( 1 - 576 T + p^{3} T^{2} \) |
| 61 | \( 1 - 74 T + p^{3} T^{2} \) |
| 67 | \( 1 + 38 T + p^{3} T^{2} \) |
| 71 | \( 1 + 456 T + p^{3} T^{2} \) |
| 73 | \( 1 + 682 T + p^{3} T^{2} \) |
| 79 | \( 1 + 704 T + p^{3} T^{2} \) |
| 83 | \( 1 + 888 T + p^{3} T^{2} \) |
| 89 | \( 1 - 1020 T + p^{3} T^{2} \) |
| 97 | \( 1 - 110 T + p^{3} T^{2} \) |
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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
−8.408276258077860413250066355506, −7.84215431106788128277401360632, −6.76539588561211774353821736286, −5.96758578413148458730033457093, −5.42818413567006078392079272763, −4.44507220079831590941310769497, −3.26984818788917386988645997618, −2.34580692349953617349608104493, −1.42889101927922347181134518969, 0,
1.42889101927922347181134518969, 2.34580692349953617349608104493, 3.26984818788917386988645997618, 4.44507220079831590941310769497, 5.42818413567006078392079272763, 5.96758578413148458730033457093, 6.76539588561211774353821736286, 7.84215431106788128277401360632, 8.408276258077860413250066355506