| L(s) = 1 | + 2-s − 3-s + 4-s + 3·5-s − 6-s + 7-s + 8-s + 9-s + 3·10-s + 5·11-s − 12-s + 14-s − 3·15-s + 16-s + 18-s + 6·19-s + 3·20-s − 21-s + 5·22-s − 2·23-s − 24-s + 4·25-s − 27-s + 28-s − 9·29-s − 3·30-s − 3·31-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s + 1.34·5-s − 0.408·6-s + 0.377·7-s + 0.353·8-s + 1/3·9-s + 0.948·10-s + 1.50·11-s − 0.288·12-s + 0.267·14-s − 0.774·15-s + 1/4·16-s + 0.235·18-s + 1.37·19-s + 0.670·20-s − 0.218·21-s + 1.06·22-s − 0.417·23-s − 0.204·24-s + 4/5·25-s − 0.192·27-s + 0.188·28-s − 1.67·29-s − 0.547·30-s − 0.538·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12138 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12138 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.880298161\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.880298161\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 17 | \( 1 \) |
| good | 5 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 + 3 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 + 13 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + T + p T^{2} \) |
| 79 | \( 1 - 13 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 4 T + p T^{2} \) |
| 97 | \( 1 - 5 T + p T^{2} \) |
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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
−16.54319608287649, −15.83509878208387, −14.97597679986908, −14.63054130962959, −13.92064508784293, −13.70640080591364, −13.00628555667907, −12.33587274405719, −11.86475253566456, −11.20173791087306, −10.85030884235309, −9.900934415458711, −9.433412080853010, −9.103501451467212, −7.907629387538645, −7.297530655682915, −6.574353694083843, −6.033471662607876, −5.545415320064477, −5.008942881013054, −4.089517990678587, −3.542003426650592, −2.401935871280809, −1.691864205074303, −1.016386962483350,
1.016386962483350, 1.691864205074303, 2.401935871280809, 3.542003426650592, 4.089517990678587, 5.008942881013054, 5.545415320064477, 6.033471662607876, 6.574353694083843, 7.297530655682915, 7.907629387538645, 9.103501451467212, 9.433412080853010, 9.900934415458711, 10.85030884235309, 11.20173791087306, 11.86475253566456, 12.33587274405719, 13.00628555667907, 13.70640080591364, 13.92064508784293, 14.63054130962959, 14.97597679986908, 15.83509878208387, 16.54319608287649
