| L(s) = 1 | + 1.73·2-s − 3-s + 0.999·4-s − 1.73·6-s + 3.46·7-s − 1.73·8-s + 9-s + 3.46·11-s − 0.999·12-s + 5.99·14-s − 5·16-s + 6·17-s + 1.73·18-s + 3.46·19-s − 3.46·21-s + 5.99·22-s + 1.73·24-s − 5·25-s − 27-s + 3.46·28-s + 6·29-s − 3.46·31-s − 5.19·32-s − 3.46·33-s + 10.3·34-s + 0.999·36-s − 6.92·37-s + ⋯ |
| L(s) = 1 | + 1.22·2-s − 0.577·3-s + 0.499·4-s − 0.707·6-s + 1.30·7-s − 0.612·8-s + 0.333·9-s + 1.04·11-s − 0.288·12-s + 1.60·14-s − 1.25·16-s + 1.45·17-s + 0.408·18-s + 0.794·19-s − 0.755·21-s + 1.27·22-s + 0.353·24-s − 25-s − 0.192·27-s + 0.654·28-s + 1.11·29-s − 0.622·31-s − 0.918·32-s − 0.603·33-s + 1.78·34-s + 0.166·36-s − 1.13·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.425937495\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.425937495\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 1.73T + 2T^{2} \) |
| 5 | \( 1 + 5T^{2} \) |
| 7 | \( 1 - 3.46T + 7T^{2} \) |
| 11 | \( 1 - 3.46T + 11T^{2} \) |
| 17 | \( 1 - 6T + 17T^{2} \) |
| 19 | \( 1 - 3.46T + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 + 3.46T + 31T^{2} \) |
| 37 | \( 1 + 6.92T + 37T^{2} \) |
| 41 | \( 1 + 6.92T + 41T^{2} \) |
| 43 | \( 1 - 4T + 43T^{2} \) |
| 47 | \( 1 + 3.46T + 47T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 + 10.3T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 + 10.3T + 67T^{2} \) |
| 71 | \( 1 - 3.46T + 71T^{2} \) |
| 73 | \( 1 + 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 + 3.46T + 83T^{2} \) |
| 89 | \( 1 - 6.92T + 89T^{2} \) |
| 97 | \( 1 + 13.8T + 97T^{2} \) |
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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
−11.36115687945316413102513683604, −10.23186052604118683807518466314, −9.173280372302837284479603894925, −8.069517973257505027287831482119, −7.01876676244882275910853449372, −5.88092650763199403169733671603, −5.21583610170390546783438457701, −4.35300108571061840677486504793, −3.34537566354701850612787093594, −1.50232947470805817130888796036,
1.50232947470805817130888796036, 3.34537566354701850612787093594, 4.35300108571061840677486504793, 5.21583610170390546783438457701, 5.88092650763199403169733671603, 7.01876676244882275910853449372, 8.069517973257505027287831482119, 9.173280372302837284479603894925, 10.23186052604118683807518466314, 11.36115687945316413102513683604
