| L(s) = 1 | + 4·5-s − 27·9-s + 92·13-s + 104·17-s − 109·25-s + 130·29-s − 214·37-s − 472·41-s − 108·45-s + 518·53-s + 468·61-s + 368·65-s − 592·73-s + 729·81-s + 416·85-s + 176·89-s + 1.81e3·97-s + 1.94e3·101-s + 1.74e3·109-s + 2.00e3·113-s − 2.48e3·117-s + ⋯ |
| L(s) = 1 | + 0.357·5-s − 9-s + 1.96·13-s + 1.48·17-s − 0.871·25-s + 0.832·29-s − 0.950·37-s − 1.79·41-s − 0.357·45-s + 1.34·53-s + 0.982·61-s + 0.702·65-s − 0.949·73-s + 81-s + 0.530·85-s + 0.209·89-s + 1.90·97-s + 1.91·101-s + 1.53·109-s + 1.66·113-s − 1.96·117-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.424447171\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.424447171\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + p^{3} T^{2} \) |
| 5 | \( 1 - 4 T + p^{3} T^{2} \) |
| 11 | \( 1 + p^{3} T^{2} \) |
| 13 | \( 1 - 92 T + p^{3} T^{2} \) |
| 17 | \( 1 - 104 T + p^{3} T^{2} \) |
| 19 | \( 1 + p^{3} T^{2} \) |
| 23 | \( 1 + p^{3} T^{2} \) |
| 29 | \( 1 - 130 T + p^{3} T^{2} \) |
| 31 | \( 1 + p^{3} T^{2} \) |
| 37 | \( 1 + 214 T + p^{3} T^{2} \) |
| 41 | \( 1 + 472 T + p^{3} T^{2} \) |
| 43 | \( 1 + p^{3} T^{2} \) |
| 47 | \( 1 + p^{3} T^{2} \) |
| 53 | \( 1 - 518 T + p^{3} T^{2} \) |
| 59 | \( 1 + p^{3} T^{2} \) |
| 61 | \( 1 - 468 T + p^{3} T^{2} \) |
| 67 | \( 1 + p^{3} T^{2} \) |
| 71 | \( 1 + p^{3} T^{2} \) |
| 73 | \( 1 + 592 T + p^{3} T^{2} \) |
| 79 | \( 1 + p^{3} T^{2} \) |
| 83 | \( 1 + p^{3} T^{2} \) |
| 89 | \( 1 - 176 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1816 T + p^{3} T^{2} \) |
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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
−8.760937786278018870297892449553, −8.530185813372817993393673349008, −7.56497469595430013294420224962, −6.41950883882780472690571684166, −5.84417270923805319755860554916, −5.14927161308044379776102294335, −3.74116557230183007481978583048, −3.17633248777438918228799808125, −1.83834121003469284607054497687, −0.77680301235398175923932901383,
0.77680301235398175923932901383, 1.83834121003469284607054497687, 3.17633248777438918228799808125, 3.74116557230183007481978583048, 5.14927161308044379776102294335, 5.84417270923805319755860554916, 6.41950883882780472690571684166, 7.56497469595430013294420224962, 8.530185813372817993393673349008, 8.760937786278018870297892449553
