| L(s) = 1 | − 2-s + 4-s − 5-s + 2·7-s − 8-s + 10-s + 3.46·11-s + 5.46·13-s − 2·14-s + 16-s + 3.46·17-s + 19-s − 20-s − 3.46·22-s − 6.92·23-s + 25-s − 5.46·26-s + 2·28-s − 3.46·29-s − 1.46·31-s − 32-s − 3.46·34-s − 2·35-s − 1.46·37-s − 38-s + 40-s + 5.46·43-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s − 0.447·5-s + 0.755·7-s − 0.353·8-s + 0.316·10-s + 1.04·11-s + 1.51·13-s − 0.534·14-s + 0.250·16-s + 0.840·17-s + 0.229·19-s − 0.223·20-s − 0.738·22-s − 1.44·23-s + 0.200·25-s − 1.07·26-s + 0.377·28-s − 0.643·29-s − 0.262·31-s − 0.176·32-s − 0.594·34-s − 0.338·35-s − 0.240·37-s − 0.162·38-s + 0.158·40-s + 0.833·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.436788697\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.436788697\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| good | 7 | \( 1 - 2T + 7T^{2} \) |
| 11 | \( 1 - 3.46T + 11T^{2} \) |
| 13 | \( 1 - 5.46T + 13T^{2} \) |
| 17 | \( 1 - 3.46T + 17T^{2} \) |
| 23 | \( 1 + 6.92T + 23T^{2} \) |
| 29 | \( 1 + 3.46T + 29T^{2} \) |
| 31 | \( 1 + 1.46T + 31T^{2} \) |
| 37 | \( 1 + 1.46T + 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 - 5.46T + 43T^{2} \) |
| 47 | \( 1 + 6.92T + 47T^{2} \) |
| 53 | \( 1 - 12.9T + 53T^{2} \) |
| 59 | \( 1 - 3.46T + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 - 14.9T + 67T^{2} \) |
| 71 | \( 1 - 6.92T + 71T^{2} \) |
| 73 | \( 1 + 4.92T + 73T^{2} \) |
| 79 | \( 1 + 1.46T + 79T^{2} \) |
| 83 | \( 1 - 2.53T + 83T^{2} \) |
| 89 | \( 1 + 6.92T + 89T^{2} \) |
| 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
−9.241854805346707690681543017457, −8.420211905976415459634058901269, −8.005260357451801954784885836325, −7.09276745297339139029493623442, −6.21256684801199713013315018218, −5.43539598025971572484151589492, −4.07741117027815684639030057273, −3.50936865721407025546104269747, −1.92858083848600256143713315087, −0.992745868829675642913465707037,
0.992745868829675642913465707037, 1.92858083848600256143713315087, 3.50936865721407025546104269747, 4.07741117027815684639030057273, 5.43539598025971572484151589492, 6.21256684801199713013315018218, 7.09276745297339139029493623442, 8.005260357451801954784885836325, 8.420211905976415459634058901269, 9.241854805346707690681543017457
