| L(s) = 1 | + 1.94·2-s + 3-s + 1.79·4-s + 1.40·5-s + 1.94·6-s − 0.394·8-s + 9-s + 2.73·10-s + 4.53·11-s + 1.79·12-s + 2.35·13-s + 1.40·15-s − 4.36·16-s + 1.10·17-s + 1.94·18-s − 0.638·19-s + 2.51·20-s + 8.83·22-s − 23-s − 0.394·24-s − 3.03·25-s + 4.58·26-s + 27-s + 2.63·29-s + 2.73·30-s + 5.19·31-s − 7.71·32-s + ⋯ |
| L(s) = 1 | + 1.37·2-s + 0.577·3-s + 0.898·4-s + 0.626·5-s + 0.795·6-s − 0.139·8-s + 0.333·9-s + 0.863·10-s + 1.36·11-s + 0.518·12-s + 0.653·13-s + 0.361·15-s − 1.09·16-s + 0.268·17-s + 0.459·18-s − 0.146·19-s + 0.563·20-s + 1.88·22-s − 0.208·23-s − 0.0805·24-s − 0.607·25-s + 0.900·26-s + 0.192·27-s + 0.488·29-s + 0.498·30-s + 0.932·31-s − 1.36·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.867402350\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.867402350\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 7 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 2 | \( 1 - 1.94T + 2T^{2} \) |
| 5 | \( 1 - 1.40T + 5T^{2} \) |
| 11 | \( 1 - 4.53T + 11T^{2} \) |
| 13 | \( 1 - 2.35T + 13T^{2} \) |
| 17 | \( 1 - 1.10T + 17T^{2} \) |
| 19 | \( 1 + 0.638T + 19T^{2} \) |
| 29 | \( 1 - 2.63T + 29T^{2} \) |
| 31 | \( 1 - 5.19T + 31T^{2} \) |
| 37 | \( 1 - 4.53T + 37T^{2} \) |
| 41 | \( 1 - 1.45T + 41T^{2} \) |
| 43 | \( 1 - 1.49T + 43T^{2} \) |
| 47 | \( 1 - 6.43T + 47T^{2} \) |
| 53 | \( 1 + 11.1T + 53T^{2} \) |
| 59 | \( 1 - 6.20T + 59T^{2} \) |
| 61 | \( 1 - 0.821T + 61T^{2} \) |
| 67 | \( 1 + 3.33T + 67T^{2} \) |
| 71 | \( 1 + 14.4T + 71T^{2} \) |
| 73 | \( 1 - 1.99T + 73T^{2} \) |
| 79 | \( 1 - 4.18T + 79T^{2} \) |
| 83 | \( 1 + 8.69T + 83T^{2} \) |
| 89 | \( 1 - 7.56T + 89T^{2} \) |
| 97 | \( 1 - 9.73T + 97T^{2} \) |
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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
−8.711858506813276667968887739851, −7.80354979818424798700162128362, −6.74157155859574026117847113926, −6.22064548278626033815330708962, −5.64087570738240423068561144641, −4.54092967350725638346828753111, −4.02152512632918270793613032709, −3.23437454790296182764243112817, −2.35380704542449241040928377849, −1.29219103340161047862692660881,
1.29219103340161047862692660881, 2.35380704542449241040928377849, 3.23437454790296182764243112817, 4.02152512632918270793613032709, 4.54092967350725638346828753111, 5.64087570738240423068561144641, 6.22064548278626033815330708962, 6.74157155859574026117847113926, 7.80354979818424798700162128362, 8.711858506813276667968887739851
