| L(s) = 1 | − 2-s + 4-s − 2.85·5-s + 4.61·7-s − 8-s + 2.85·10-s − 3.23·13-s − 4.61·14-s + 16-s + 2.47·17-s − 3.23·19-s − 2.85·20-s + 3.23·23-s + 3.14·25-s + 3.23·26-s + 4.61·28-s + 0.381·29-s + 8.61·31-s − 32-s − 2.47·34-s − 13.1·35-s + 1.52·37-s + 3.23·38-s + 2.85·40-s + 3.23·41-s − 3.23·43-s − 3.23·46-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s − 1.27·5-s + 1.74·7-s − 0.353·8-s + 0.902·10-s − 0.897·13-s − 1.23·14-s + 0.250·16-s + 0.599·17-s − 0.742·19-s − 0.638·20-s + 0.674·23-s + 0.629·25-s + 0.634·26-s + 0.872·28-s + 0.0709·29-s + 1.54·31-s − 0.176·32-s − 0.423·34-s − 2.22·35-s + 0.251·37-s + 0.524·38-s + 0.451·40-s + 0.505·41-s − 0.493·43-s − 0.477·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2178 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2178 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.121720988\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.121720988\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 11 | \( 1 \) |
| good | 5 | \( 1 + 2.85T + 5T^{2} \) |
| 7 | \( 1 - 4.61T + 7T^{2} \) |
| 13 | \( 1 + 3.23T + 13T^{2} \) |
| 17 | \( 1 - 2.47T + 17T^{2} \) |
| 19 | \( 1 + 3.23T + 19T^{2} \) |
| 23 | \( 1 - 3.23T + 23T^{2} \) |
| 29 | \( 1 - 0.381T + 29T^{2} \) |
| 31 | \( 1 - 8.61T + 31T^{2} \) |
| 37 | \( 1 - 1.52T + 37T^{2} \) |
| 41 | \( 1 - 3.23T + 41T^{2} \) |
| 43 | \( 1 + 3.23T + 43T^{2} \) |
| 47 | \( 1 + 2.47T + 47T^{2} \) |
| 53 | \( 1 - 3.85T + 53T^{2} \) |
| 59 | \( 1 - 3.85T + 59T^{2} \) |
| 61 | \( 1 + 6.76T + 61T^{2} \) |
| 67 | \( 1 - 4T + 67T^{2} \) |
| 71 | \( 1 + 12.4T + 71T^{2} \) |
| 73 | \( 1 + 11.3T + 73T^{2} \) |
| 79 | \( 1 + 9.32T + 79T^{2} \) |
| 83 | \( 1 - 0.673T + 83T^{2} \) |
| 89 | \( 1 - 11.7T + 89T^{2} \) |
| 97 | \( 1 - 11.3T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.668587266010774136304825300989, −8.366708328881274068351563105091, −7.54330300621626421908873620274, −7.26989882183471939330900259141, −5.98523292275059972433830180783, −4.80297883282914629164846364899, −4.41424157457398053370909750635, −3.12864324384724769739190711590, −1.99330556163395116428919568696, −0.78354054822630437297368677856,
0.78354054822630437297368677856, 1.99330556163395116428919568696, 3.12864324384724769739190711590, 4.41424157457398053370909750635, 4.80297883282914629164846364899, 5.98523292275059972433830180783, 7.26989882183471939330900259141, 7.54330300621626421908873620274, 8.366708328881274068351563105091, 8.668587266010774136304825300989
