| L(s) = 1 | + 3-s + 2.68·5-s − 0.168·7-s + 9-s − 5.32·11-s + 3.84·13-s + 2.68·15-s − 6.65·17-s − 6.89·19-s − 0.168·21-s + 2.22·25-s + 27-s − 0.569·29-s − 6.95·31-s − 5.32·33-s − 0.452·35-s + 8.85·37-s + 3.84·39-s + 4.86·41-s − 6.75·43-s + 2.68·45-s − 2.26·47-s − 6.97·49-s − 6.65·51-s + 4.87·53-s − 14.3·55-s − 6.89·57-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.20·5-s − 0.0636·7-s + 0.333·9-s − 1.60·11-s + 1.06·13-s + 0.693·15-s − 1.61·17-s − 1.58·19-s − 0.0367·21-s + 0.444·25-s + 0.192·27-s − 0.105·29-s − 1.24·31-s − 0.926·33-s − 0.0765·35-s + 1.45·37-s + 0.615·39-s + 0.760·41-s − 1.03·43-s + 0.400·45-s − 0.330·47-s − 0.995·49-s − 0.932·51-s + 0.669·53-s − 1.92·55-s − 0.912·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6348 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6348 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 23 | \( 1 \) |
| good | 5 | \( 1 - 2.68T + 5T^{2} \) |
| 7 | \( 1 + 0.168T + 7T^{2} \) |
| 11 | \( 1 + 5.32T + 11T^{2} \) |
| 13 | \( 1 - 3.84T + 13T^{2} \) |
| 17 | \( 1 + 6.65T + 17T^{2} \) |
| 19 | \( 1 + 6.89T + 19T^{2} \) |
| 29 | \( 1 + 0.569T + 29T^{2} \) |
| 31 | \( 1 + 6.95T + 31T^{2} \) |
| 37 | \( 1 - 8.85T + 37T^{2} \) |
| 41 | \( 1 - 4.86T + 41T^{2} \) |
| 43 | \( 1 + 6.75T + 43T^{2} \) |
| 47 | \( 1 + 2.26T + 47T^{2} \) |
| 53 | \( 1 - 4.87T + 53T^{2} \) |
| 59 | \( 1 - 2.28T + 59T^{2} \) |
| 61 | \( 1 + 13.9T + 61T^{2} \) |
| 67 | \( 1 + 1.61T + 67T^{2} \) |
| 71 | \( 1 + 9.49T + 71T^{2} \) |
| 73 | \( 1 - 3.83T + 73T^{2} \) |
| 79 | \( 1 - 2.62T + 79T^{2} \) |
| 83 | \( 1 + 8.96T + 83T^{2} \) |
| 89 | \( 1 - 6.28T + 89T^{2} \) |
| 97 | \( 1 + 13.2T + 97T^{2} \) |
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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
−7.84360264033410969958046470605, −6.88397391061607388678339893018, −6.21041426156051016967268602578, −5.69643509266212131646575713201, −4.75405374473572423211861216971, −4.07721895063887129711942426370, −2.94877417250744205519665496930, −2.27258763834649460841396064851, −1.68361765014624122870581937833, 0,
1.68361765014624122870581937833, 2.27258763834649460841396064851, 2.94877417250744205519665496930, 4.07721895063887129711942426370, 4.75405374473572423211861216971, 5.69643509266212131646575713201, 6.21041426156051016967268602578, 6.88397391061607388678339893018, 7.84360264033410969958046470605
