| L(s) = 1 | − 3.09·2-s + 1.60·4-s + 5·5-s + 19.8·8-s − 15.4·10-s + 31.1·11-s + 54.1·13-s − 74.2·16-s − 81.8·17-s − 37.3·19-s + 8.03·20-s − 96.6·22-s − 116.·23-s + 25·25-s − 167.·26-s − 22.5·29-s + 89.9·31-s + 71.6·32-s + 253.·34-s + 344.·37-s + 115.·38-s + 99.0·40-s + 245.·41-s + 41.4·43-s + 50.1·44-s + 360.·46-s + 431.·47-s + ⋯ |
| L(s) = 1 | − 1.09·2-s + 0.200·4-s + 0.447·5-s + 0.875·8-s − 0.490·10-s + 0.854·11-s + 1.15·13-s − 1.16·16-s − 1.16·17-s − 0.451·19-s + 0.0898·20-s − 0.936·22-s − 1.05·23-s + 0.200·25-s − 1.26·26-s − 0.144·29-s + 0.521·31-s + 0.395·32-s + 1.28·34-s + 1.53·37-s + 0.494·38-s + 0.391·40-s + 0.936·41-s + 0.147·43-s + 0.171·44-s + 1.15·46-s + 1.33·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.276795707\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.276795707\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - 5T \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + 3.09T + 8T^{2} \) |
| 11 | \( 1 - 31.1T + 1.33e3T^{2} \) |
| 13 | \( 1 - 54.1T + 2.19e3T^{2} \) |
| 17 | \( 1 + 81.8T + 4.91e3T^{2} \) |
| 19 | \( 1 + 37.3T + 6.85e3T^{2} \) |
| 23 | \( 1 + 116.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 22.5T + 2.43e4T^{2} \) |
| 31 | \( 1 - 89.9T + 2.97e4T^{2} \) |
| 37 | \( 1 - 344.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 245.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 41.4T + 7.95e4T^{2} \) |
| 47 | \( 1 - 431.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 263.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 627.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 715.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 809.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 54.8T + 3.57e5T^{2} \) |
| 73 | \( 1 + 508.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 61.8T + 4.93e5T^{2} \) |
| 83 | \( 1 - 560.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 102.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 253.T + 9.12e5T^{2} \) |
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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
−8.874358988116054479859372795718, −8.167470345434317015361956317666, −7.31787704947732384078709545889, −6.39788241576888760184819297090, −5.87053191887275755639781955626, −4.44885908598209440488038566098, −3.99628107289914328880987689160, −2.46625098880710676226692208646, −1.54008651071844961356401458186, −0.64507118477363418421099294959,
0.64507118477363418421099294959, 1.54008651071844961356401458186, 2.46625098880710676226692208646, 3.99628107289914328880987689160, 4.44885908598209440488038566098, 5.87053191887275755639781955626, 6.39788241576888760184819297090, 7.31787704947732384078709545889, 8.167470345434317015361956317666, 8.874358988116054479859372795718
