| L(s) = 1 | + 4.38·2-s − 3·3-s + 11.2·4-s + 5·5-s − 13.1·6-s + 11.7·7-s + 14.2·8-s + 9·9-s + 21.9·10-s − 33.7·12-s + 72.8·13-s + 51.4·14-s − 15·15-s − 27.3·16-s + 9.89·17-s + 39.4·18-s − 0.0238·19-s + 56.2·20-s − 35.1·21-s + 73.0·23-s − 42.8·24-s + 25·25-s + 319.·26-s − 27·27-s + 132.·28-s − 202.·29-s − 65.8·30-s + ⋯ |
| L(s) = 1 | + 1.55·2-s − 0.577·3-s + 1.40·4-s + 0.447·5-s − 0.895·6-s + 0.633·7-s + 0.631·8-s + 0.333·9-s + 0.693·10-s − 0.812·12-s + 1.55·13-s + 0.982·14-s − 0.258·15-s − 0.426·16-s + 0.141·17-s + 0.517·18-s − 0.000288·19-s + 0.629·20-s − 0.365·21-s + 0.662·23-s − 0.364·24-s + 0.200·25-s + 2.41·26-s − 0.192·27-s + 0.891·28-s − 1.29·29-s − 0.400·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(5.988254464\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.988254464\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 3T \) |
| 5 | \( 1 - 5T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 4.38T + 8T^{2} \) |
| 7 | \( 1 - 11.7T + 343T^{2} \) |
| 13 | \( 1 - 72.8T + 2.19e3T^{2} \) |
| 17 | \( 1 - 9.89T + 4.91e3T^{2} \) |
| 19 | \( 1 + 0.0238T + 6.85e3T^{2} \) |
| 23 | \( 1 - 73.0T + 1.21e4T^{2} \) |
| 29 | \( 1 + 202.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 181.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 299.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 88.5T + 6.89e4T^{2} \) |
| 43 | \( 1 - 146.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 185.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 347.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 691.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 491.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 715.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 541.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 159.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 212.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 413.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.09e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 567.T + 9.12e5T^{2} \) |
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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
−8.888001193984314071516761875967, −7.951989099210186422486139458842, −6.89221369070805534834107901446, −6.15430394062673646371756207787, −5.63811466363142732580044960475, −4.85610997158224260717376825064, −4.09646731509594877435047184968, −3.23530475626282092800935154425, −2.06958092893821439889098846596, −0.988507701208441634681812048905,
0.988507701208441634681812048905, 2.06958092893821439889098846596, 3.23530475626282092800935154425, 4.09646731509594877435047184968, 4.85610997158224260717376825064, 5.63811466363142732580044960475, 6.15430394062673646371756207787, 6.89221369070805534834107901446, 7.951989099210186422486139458842, 8.888001193984314071516761875967
