| L(s) = 1 | − 2.52·2-s + 3-s + 4.39·4-s − 2.52·6-s + 7-s − 6.05·8-s + 9-s + 11-s + 4.39·12-s − 0.133·13-s − 2.52·14-s + 6.52·16-s + 5.05·17-s − 2.52·18-s − 0.924·19-s + 21-s − 2.52·22-s + 7.05·23-s − 6.05·24-s + 0.337·26-s + 27-s + 4.39·28-s + 3.86·29-s + 2.79·31-s − 4.39·32-s + 33-s − 12.7·34-s + ⋯ |
| L(s) = 1 | − 1.78·2-s + 0.577·3-s + 2.19·4-s − 1.03·6-s + 0.377·7-s − 2.14·8-s + 0.333·9-s + 0.301·11-s + 1.26·12-s − 0.0370·13-s − 0.675·14-s + 1.63·16-s + 1.22·17-s − 0.596·18-s − 0.212·19-s + 0.218·21-s − 0.539·22-s + 1.47·23-s − 1.23·24-s + 0.0662·26-s + 0.192·27-s + 0.830·28-s + 0.717·29-s + 0.501·31-s − 0.777·32-s + 0.174·33-s − 2.19·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5775 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5775 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.297867369\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.297867369\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| good | 2 | \( 1 + 2.52T + 2T^{2} \) |
| 13 | \( 1 + 0.133T + 13T^{2} \) |
| 17 | \( 1 - 5.05T + 17T^{2} \) |
| 19 | \( 1 + 0.924T + 19T^{2} \) |
| 23 | \( 1 - 7.05T + 23T^{2} \) |
| 29 | \( 1 - 3.86T + 29T^{2} \) |
| 31 | \( 1 - 2.79T + 31T^{2} \) |
| 37 | \( 1 + 9.98T + 37T^{2} \) |
| 41 | \( 1 - 11.8T + 41T^{2} \) |
| 43 | \( 1 - 3.05T + 43T^{2} \) |
| 47 | \( 1 - 3.07T + 47T^{2} \) |
| 53 | \( 1 - 4.79T + 53T^{2} \) |
| 59 | \( 1 + 12.6T + 59T^{2} \) |
| 61 | \( 1 - 6T + 61T^{2} \) |
| 67 | \( 1 + 8.92T + 67T^{2} \) |
| 71 | \( 1 + 6.11T + 71T^{2} \) |
| 73 | \( 1 + 7.86T + 73T^{2} \) |
| 79 | \( 1 - 14.1T + 79T^{2} \) |
| 83 | \( 1 - 1.20T + 83T^{2} \) |
| 89 | \( 1 - 15.5T + 89T^{2} \) |
| 97 | \( 1 + 12.7T + 97T^{2} \) |
| 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
−8.191087861656377565777475769716, −7.61555172767309893398561553692, −7.10612926901388650389307285778, −6.35126520861930082655378658740, −5.42177486039878615032883217843, −4.37835653654235994495557250067, −3.22809562757261990545261078968, −2.55816778016034016102384711146, −1.52380017699210907480318738253, −0.838017382016237135363778721970,
0.838017382016237135363778721970, 1.52380017699210907480318738253, 2.55816778016034016102384711146, 3.22809562757261990545261078968, 4.37835653654235994495557250067, 5.42177486039878615032883217843, 6.35126520861930082655378658740, 7.10612926901388650389307285778, 7.61555172767309893398561553692, 8.191087861656377565777475769716
