| L(s) = 1 | − 2.51·2-s − 1.27·3-s + 4.34·4-s + 3.65·5-s + 3.21·6-s + 1.49·7-s − 5.90·8-s − 1.37·9-s − 9.19·10-s − 5.53·12-s + 13-s − 3.75·14-s − 4.65·15-s + 6.18·16-s − 4.13·17-s + 3.46·18-s − 5.60·19-s + 15.8·20-s − 1.90·21-s + 5.79·23-s + 7.53·24-s + 8.32·25-s − 2.51·26-s + 5.57·27-s + 6.48·28-s + 3.41·29-s + 11.7·30-s + ⋯ |
| L(s) = 1 | − 1.78·2-s − 0.736·3-s + 2.17·4-s + 1.63·5-s + 1.31·6-s + 0.564·7-s − 2.08·8-s − 0.458·9-s − 2.90·10-s − 1.59·12-s + 0.277·13-s − 1.00·14-s − 1.20·15-s + 1.54·16-s − 1.00·17-s + 0.815·18-s − 1.28·19-s + 3.54·20-s − 0.415·21-s + 1.20·23-s + 1.53·24-s + 1.66·25-s − 0.493·26-s + 1.07·27-s + 1.22·28-s + 0.633·29-s + 2.14·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1573 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1573 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7522614608\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7522614608\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 2 | \( 1 + 2.51T + 2T^{2} \) |
| 3 | \( 1 + 1.27T + 3T^{2} \) |
| 5 | \( 1 - 3.65T + 5T^{2} \) |
| 7 | \( 1 - 1.49T + 7T^{2} \) |
| 17 | \( 1 + 4.13T + 17T^{2} \) |
| 19 | \( 1 + 5.60T + 19T^{2} \) |
| 23 | \( 1 - 5.79T + 23T^{2} \) |
| 29 | \( 1 - 3.41T + 29T^{2} \) |
| 31 | \( 1 - 4.97T + 31T^{2} \) |
| 37 | \( 1 + 7.75T + 37T^{2} \) |
| 41 | \( 1 - 5.89T + 41T^{2} \) |
| 43 | \( 1 - 4.53T + 43T^{2} \) |
| 47 | \( 1 + 0.0839T + 47T^{2} \) |
| 53 | \( 1 - 6.53T + 53T^{2} \) |
| 59 | \( 1 - 6.56T + 59T^{2} \) |
| 61 | \( 1 - 0.135T + 61T^{2} \) |
| 67 | \( 1 - 3.41T + 67T^{2} \) |
| 71 | \( 1 + 9.88T + 71T^{2} \) |
| 73 | \( 1 - 12.3T + 73T^{2} \) |
| 79 | \( 1 - 3.52T + 79T^{2} \) |
| 83 | \( 1 - 15.8T + 83T^{2} \) |
| 89 | \( 1 + 3.76T + 89T^{2} \) |
| 97 | \( 1 + 16.7T + 97T^{2} \) |
| 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
−9.306340643125634182663411189703, −8.784485507695526154689332284484, −8.214501113077209332410548256022, −6.84707366944860611302687826986, −6.48540101882945697637312137465, −5.68181347029636439328981539273, −4.74257503042412875795892282432, −2.66231840884883210895723760408, −1.94898239303517474193623356113, −0.833420034208976337728248719394,
0.833420034208976337728248719394, 1.94898239303517474193623356113, 2.66231840884883210895723760408, 4.74257503042412875795892282432, 5.68181347029636439328981539273, 6.48540101882945697637312137465, 6.84707366944860611302687826986, 8.214501113077209332410548256022, 8.784485507695526154689332284484, 9.306340643125634182663411189703
