| L(s) = 1 | − 11-s − 2·13-s − 4·17-s + 2·19-s + 4·29-s − 2·31-s + 37-s − 10·41-s + 2·43-s − 8·47-s − 7·49-s + 10·53-s − 10·61-s + 8·67-s − 16·71-s − 10·73-s − 14·79-s + 4·83-s + 4·89-s + 2·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | − 0.301·11-s − 0.554·13-s − 0.970·17-s + 0.458·19-s + 0.742·29-s − 0.359·31-s + 0.164·37-s − 1.56·41-s + 0.304·43-s − 1.16·47-s − 49-s + 1.37·53-s − 1.28·61-s + 0.977·67-s − 1.89·71-s − 1.17·73-s − 1.57·79-s + 0.439·83-s + 0.423·89-s + 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 366300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 366300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4955503429\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4955503429\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 11 | \( 1 + T \) |
| 37 | \( 1 - T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 4 T + p T^{2} \) |
| 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
−12.49321759909413, −11.97288713768993, −11.65068754637080, −11.22638803309503, −10.67137508803032, −10.07727958679310, −10.04132616770079, −9.236870395162174, −8.933767656978213, −8.361722318304705, −8.031009530899837, −7.274982573988363, −7.134378424976255, −6.480579104125337, −6.057435688075422, −5.464365724379611, −4.905804453184274, −4.614964957112161, −4.026717062637784, −3.325801994951993, −2.915720454012373, −2.339096780380864, −1.726145769886823, −1.163249164735952, −0.1822137380482249,
0.1822137380482249, 1.163249164735952, 1.726145769886823, 2.339096780380864, 2.915720454012373, 3.325801994951993, 4.026717062637784, 4.614964957112161, 4.905804453184274, 5.464365724379611, 6.057435688075422, 6.480579104125337, 7.134378424976255, 7.274982573988363, 8.031009530899837, 8.361722318304705, 8.933767656978213, 9.236870395162174, 10.04132616770079, 10.07727958679310, 10.67137508803032, 11.22638803309503, 11.65068754637080, 11.97288713768993, 12.49321759909413
