| L(s) = 1 | − 2·2-s − 3·3-s + 4·4-s − 4.88·5-s + 6·6-s + 7·7-s − 8·8-s + 9·9-s + 9.76·10-s + 63.9·11-s − 12·12-s + 61.9·13-s − 14·14-s + 14.6·15-s + 16·16-s + 22.0·17-s − 18·18-s + 19·19-s − 19.5·20-s − 21·21-s − 127.·22-s + 22.9·23-s + 24·24-s − 101.·25-s − 123.·26-s − 27·27-s + 28·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.436·5-s + 0.408·6-s + 0.377·7-s − 0.353·8-s + 0.333·9-s + 0.308·10-s + 1.75·11-s − 0.288·12-s + 1.32·13-s − 0.267·14-s + 0.252·15-s + 0.250·16-s + 0.314·17-s − 0.235·18-s + 0.229·19-s − 0.218·20-s − 0.218·21-s − 1.23·22-s + 0.208·23-s + 0.204·24-s − 0.809·25-s − 0.934·26-s − 0.192·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 798 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 798 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.382157867\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.382157867\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 2T \) |
| 3 | \( 1 + 3T \) |
| 7 | \( 1 - 7T \) |
| 19 | \( 1 - 19T \) |
| good | 5 | \( 1 + 4.88T + 125T^{2} \) |
| 11 | \( 1 - 63.9T + 1.33e3T^{2} \) |
| 13 | \( 1 - 61.9T + 2.19e3T^{2} \) |
| 17 | \( 1 - 22.0T + 4.91e3T^{2} \) |
| 23 | \( 1 - 22.9T + 1.21e4T^{2} \) |
| 29 | \( 1 + 110.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 54.7T + 2.97e4T^{2} \) |
| 37 | \( 1 - 286.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 56.8T + 6.89e4T^{2} \) |
| 43 | \( 1 - 299.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 54.7T + 1.03e5T^{2} \) |
| 53 | \( 1 + 620.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 100.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 560.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 208.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 228.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 259.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 747.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 407.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 442.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 591.T + 9.12e5T^{2} \) |
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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
−9.738802073242857531180895491202, −9.101094060493049738780033166719, −8.210602903894603230402082181825, −7.35662488805124208318633554463, −6.41224121636780442045465147888, −5.75165981716120699794852199523, −4.29528743052201813591849008947, −3.51136711926022868146551514980, −1.68209153937716106404003647503, −0.817450478268177221619171371813,
0.817450478268177221619171371813, 1.68209153937716106404003647503, 3.51136711926022868146551514980, 4.29528743052201813591849008947, 5.75165981716120699794852199523, 6.41224121636780442045465147888, 7.35662488805124208318633554463, 8.210602903894603230402082181825, 9.101094060493049738780033166719, 9.738802073242857531180895491202
