| L(s) = 1 | + 2-s + 3-s + 4-s + 2.59·5-s + 6-s + 1.23·7-s + 8-s + 9-s + 2.59·10-s + 0.622·11-s + 12-s − 1.32·13-s + 1.23·14-s + 2.59·15-s + 16-s + 4.70·17-s + 18-s − 3.79·19-s + 2.59·20-s + 1.23·21-s + 0.622·22-s + 24-s + 1.73·25-s − 1.32·26-s + 27-s + 1.23·28-s − 1.11·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s + 1.16·5-s + 0.408·6-s + 0.467·7-s + 0.353·8-s + 0.333·9-s + 0.820·10-s + 0.187·11-s + 0.288·12-s − 0.367·13-s + 0.330·14-s + 0.669·15-s + 0.250·16-s + 1.14·17-s + 0.235·18-s − 0.869·19-s + 0.580·20-s + 0.269·21-s + 0.132·22-s + 0.204·24-s + 0.346·25-s − 0.259·26-s + 0.192·27-s + 0.233·28-s − 0.206·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3174 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3174 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.960954705\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.960954705\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 23 | \( 1 \) |
| good | 5 | \( 1 - 2.59T + 5T^{2} \) |
| 7 | \( 1 - 1.23T + 7T^{2} \) |
| 11 | \( 1 - 0.622T + 11T^{2} \) |
| 13 | \( 1 + 1.32T + 13T^{2} \) |
| 17 | \( 1 - 4.70T + 17T^{2} \) |
| 19 | \( 1 + 3.79T + 19T^{2} \) |
| 29 | \( 1 + 1.11T + 29T^{2} \) |
| 31 | \( 1 - 9.45T + 31T^{2} \) |
| 37 | \( 1 - 1.62T + 37T^{2} \) |
| 41 | \( 1 + 11.8T + 41T^{2} \) |
| 43 | \( 1 - 1.67T + 43T^{2} \) |
| 47 | \( 1 + 6.94T + 47T^{2} \) |
| 53 | \( 1 - 11.2T + 53T^{2} \) |
| 59 | \( 1 + 9.74T + 59T^{2} \) |
| 61 | \( 1 - 14.3T + 61T^{2} \) |
| 67 | \( 1 - 6.12T + 67T^{2} \) |
| 71 | \( 1 - 6.49T + 71T^{2} \) |
| 73 | \( 1 + 13.0T + 73T^{2} \) |
| 79 | \( 1 - 12.0T + 79T^{2} \) |
| 83 | \( 1 + 7.39T + 83T^{2} \) |
| 89 | \( 1 + 3.38T + 89T^{2} \) |
| 97 | \( 1 + 11.4T + 97T^{2} \) |
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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
−8.485865653277022039438029629953, −8.053004275077849273687870246863, −6.98885440980540425873096249227, −6.37575588210953844732608454170, −5.52983789271809491163755147126, −4.89259782096406418940822255942, −3.98212510192817122239883448776, −2.99009615418316014719144680010, −2.18279448857432995183954222558, −1.35804934788299145953458085143,
1.35804934788299145953458085143, 2.18279448857432995183954222558, 2.99009615418316014719144680010, 3.98212510192817122239883448776, 4.89259782096406418940822255942, 5.52983789271809491163755147126, 6.37575588210953844732608454170, 6.98885440980540425873096249227, 8.053004275077849273687870246863, 8.485865653277022039438029629953
