| L(s) = 1 | − 5-s + 3·11-s − 13-s − 2·17-s − 4·19-s + 4·23-s + 25-s + 4·29-s + 3·37-s − 2·41-s − 2·43-s − 6·47-s − 13·53-s − 3·55-s − 5·59-s − 7·61-s + 65-s − 13·67-s + 5·71-s + 11·73-s + 13·79-s − 4·83-s + 2·85-s + 89-s + 4·95-s + 2·97-s + 101-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.904·11-s − 0.277·13-s − 0.485·17-s − 0.917·19-s + 0.834·23-s + 1/5·25-s + 0.742·29-s + 0.493·37-s − 0.312·41-s − 0.304·43-s − 0.875·47-s − 1.78·53-s − 0.404·55-s − 0.650·59-s − 0.896·61-s + 0.124·65-s − 1.58·67-s + 0.593·71-s + 1.28·73-s + 1.46·79-s − 0.439·83-s + 0.216·85-s + 0.105·89-s + 0.410·95-s + 0.203·97-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 229320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 229320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 11 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 13 T + p T^{2} \) |
| 59 | \( 1 + 5 T + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 + 13 T + p T^{2} \) |
| 71 | \( 1 - 5 T + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 - 13 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−13.05115453731669, −12.70950923215912, −12.17996906786545, −11.83836585999623, −11.28409335906918, −10.86175870938227, −10.53884680335892, −9.816757376382219, −9.378562890595973, −8.953308154799152, −8.528592245103865, −7.883005670427339, −7.645327258754220, −6.773160166136555, −6.562157765035890, −6.206770433786260, −5.379247867024254, −4.745584830077514, −4.504440119000499, −3.906075483110976, −3.236394057079497, −2.870832811569076, −2.020746127537116, −1.525161810274343, −0.7484391704108139, 0,
0.7484391704108139, 1.525161810274343, 2.020746127537116, 2.870832811569076, 3.236394057079497, 3.906075483110976, 4.504440119000499, 4.745584830077514, 5.379247867024254, 6.206770433786260, 6.562157765035890, 6.773160166136555, 7.645327258754220, 7.883005670427339, 8.528592245103865, 8.953308154799152, 9.378562890595973, 9.816757376382219, 10.53884680335892, 10.86175870938227, 11.28409335906918, 11.83836585999623, 12.17996906786545, 12.70950923215912, 13.05115453731669
