| L(s) = 1 | + 11-s − 2·13-s − 8·17-s + 6·19-s − 8·23-s − 5·25-s + 6·29-s − 10·31-s + 2·37-s − 8·41-s − 4·43-s + 2·47-s − 2·53-s − 8·59-s − 2·61-s + 4·67-s + 4·71-s + 4·73-s − 8·79-s − 18·83-s + 2·89-s + 14·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | + 0.301·11-s − 0.554·13-s − 1.94·17-s + 1.37·19-s − 1.66·23-s − 25-s + 1.11·29-s − 1.79·31-s + 0.328·37-s − 1.24·41-s − 0.609·43-s + 0.291·47-s − 0.274·53-s − 1.04·59-s − 0.256·61-s + 0.488·67-s + 0.474·71-s + 0.468·73-s − 0.900·79-s − 1.97·83-s + 0.211·89-s + 1.42·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38808 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38808 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9061180112\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9061180112\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 8 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 18 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−14.81558599792455, −14.21862649924543, −13.71619891860994, −13.47832269625160, −12.64626010585505, −12.22952198735579, −11.55585689928562, −11.37086151984484, −10.57398679977997, −10.00954508190653, −9.528575305728995, −9.052740885919570, −8.363793439807874, −7.887691743558582, −7.177188418609112, −6.775654117861885, −6.068411912027722, −5.536167755499676, −4.806909180766001, −4.258081772744028, −3.647375073119619, −2.899586758364424, −2.064074367545952, −1.622611795264238, −0.3315116968229476,
0.3315116968229476, 1.622611795264238, 2.064074367545952, 2.899586758364424, 3.647375073119619, 4.258081772744028, 4.806909180766001, 5.536167755499676, 6.068411912027722, 6.775654117861885, 7.177188418609112, 7.887691743558582, 8.363793439807874, 9.052740885919570, 9.528575305728995, 10.00954508190653, 10.57398679977997, 11.37086151984484, 11.55585689928562, 12.22952198735579, 12.64626010585505, 13.47832269625160, 13.71619891860994, 14.21862649924543, 14.81558599792455
