| L(s) = 1 | + 2-s + 0.378·3-s + 4-s − 3.54·5-s + 0.378·6-s + 2.78·7-s + 8-s − 2.85·9-s − 3.54·10-s + 11-s + 0.378·12-s − 3.06·13-s + 2.78·14-s − 1.34·15-s + 16-s − 3.55·17-s − 2.85·18-s + 6.32·19-s − 3.54·20-s + 1.05·21-s + 22-s + 1.73·23-s + 0.378·24-s + 7.58·25-s − 3.06·26-s − 2.21·27-s + 2.78·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.218·3-s + 0.5·4-s − 1.58·5-s + 0.154·6-s + 1.05·7-s + 0.353·8-s − 0.952·9-s − 1.12·10-s + 0.301·11-s + 0.109·12-s − 0.851·13-s + 0.744·14-s − 0.346·15-s + 0.250·16-s − 0.861·17-s − 0.673·18-s + 1.45·19-s − 0.793·20-s + 0.230·21-s + 0.213·22-s + 0.361·23-s + 0.0773·24-s + 1.51·25-s − 0.601·26-s − 0.426·27-s + 0.526·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4334 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4334 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.382529251\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.382529251\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 197 | \( 1 - T \) |
| good | 3 | \( 1 - 0.378T + 3T^{2} \) |
| 5 | \( 1 + 3.54T + 5T^{2} \) |
| 7 | \( 1 - 2.78T + 7T^{2} \) |
| 13 | \( 1 + 3.06T + 13T^{2} \) |
| 17 | \( 1 + 3.55T + 17T^{2} \) |
| 19 | \( 1 - 6.32T + 19T^{2} \) |
| 23 | \( 1 - 1.73T + 23T^{2} \) |
| 29 | \( 1 + 3.94T + 29T^{2} \) |
| 31 | \( 1 - 8.18T + 31T^{2} \) |
| 37 | \( 1 - 8.83T + 37T^{2} \) |
| 41 | \( 1 + 8.43T + 41T^{2} \) |
| 43 | \( 1 + 3.35T + 43T^{2} \) |
| 47 | \( 1 - 7.03T + 47T^{2} \) |
| 53 | \( 1 - 11.5T + 53T^{2} \) |
| 59 | \( 1 - 3.59T + 59T^{2} \) |
| 61 | \( 1 + 7.47T + 61T^{2} \) |
| 67 | \( 1 - 8.49T + 67T^{2} \) |
| 71 | \( 1 + 9.29T + 71T^{2} \) |
| 73 | \( 1 - 12.3T + 73T^{2} \) |
| 79 | \( 1 + 4.61T + 79T^{2} \) |
| 83 | \( 1 - 9.35T + 83T^{2} \) |
| 89 | \( 1 + 7.40T + 89T^{2} \) |
| 97 | \( 1 - 18.3T + 97T^{2} \) |
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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
−8.253174927344751190313380399214, −7.56326353419091297913303031925, −7.16469606497465915817916440813, −6.10850595571524079845842468909, −5.07212516468970395054218181019, −4.69466204694893193965752545099, −3.83666100242999621128867260624, −3.09826238924124019254871031691, −2.23691201843256077363511770352, −0.76493824312640059204391595382,
0.76493824312640059204391595382, 2.23691201843256077363511770352, 3.09826238924124019254871031691, 3.83666100242999621128867260624, 4.69466204694893193965752545099, 5.07212516468970395054218181019, 6.10850595571524079845842468909, 7.16469606497465915817916440813, 7.56326353419091297913303031925, 8.253174927344751190313380399214
