| L(s) = 1 | + 2.46·2-s − 1.92·4-s − 19.2·7-s − 24.4·8-s − 39.8·11-s + 1.00·13-s − 47.4·14-s − 44.8·16-s − 52.6·17-s − 49.5·19-s − 98.1·22-s − 27.4·23-s + 2.48·26-s + 37.1·28-s + 254.·29-s − 168.·31-s + 85.2·32-s − 129.·34-s − 419.·37-s − 122.·38-s − 398.·41-s + 358.·43-s + 76.8·44-s − 67.5·46-s − 141.·47-s + 27.1·49-s − 1.94·52-s + ⋯ |
| L(s) = 1 | + 0.871·2-s − 0.241·4-s − 1.03·7-s − 1.08·8-s − 1.09·11-s + 0.0215·13-s − 0.904·14-s − 0.700·16-s − 0.750·17-s − 0.598·19-s − 0.951·22-s − 0.248·23-s + 0.0187·26-s + 0.250·28-s + 1.63·29-s − 0.977·31-s + 0.470·32-s − 0.653·34-s − 1.86·37-s − 0.521·38-s − 1.51·41-s + 1.27·43-s + 0.263·44-s − 0.216·46-s − 0.438·47-s + 0.0792·49-s − 0.00519·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.8969400126\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8969400126\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 - 2.46T + 8T^{2} \) |
| 7 | \( 1 + 19.2T + 343T^{2} \) |
| 11 | \( 1 + 39.8T + 1.33e3T^{2} \) |
| 13 | \( 1 - 1.00T + 2.19e3T^{2} \) |
| 17 | \( 1 + 52.6T + 4.91e3T^{2} \) |
| 19 | \( 1 + 49.5T + 6.85e3T^{2} \) |
| 23 | \( 1 + 27.4T + 1.21e4T^{2} \) |
| 29 | \( 1 - 254.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 168.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 419.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 398.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 358.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 141.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 290.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 28.7T + 2.05e5T^{2} \) |
| 61 | \( 1 - 732.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 176.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 802.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 512.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 612.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 80.8T + 5.71e5T^{2} \) |
| 89 | \( 1 + 24.0T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.36e3T + 9.12e5T^{2} \) |
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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
−8.755569153661944103911573089267, −8.138718474552600347567432311598, −6.87089959425777367643913929939, −6.39200250982845776569747462204, −5.41423869969918333935702842071, −4.82088909100750546660161637078, −3.82044137263054038076152027989, −3.11363461417654064807799464756, −2.20381355296121797988925981569, −0.35588066229596096729088267515,
0.35588066229596096729088267515, 2.20381355296121797988925981569, 3.11363461417654064807799464756, 3.82044137263054038076152027989, 4.82088909100750546660161637078, 5.41423869969918333935702842071, 6.39200250982845776569747462204, 6.87089959425777367643913929939, 8.138718474552600347567432311598, 8.755569153661944103911573089267
