| L(s) = 1 | − 2-s + 0.867·3-s + 4-s − 3.43·5-s − 0.867·6-s − 2.50·7-s − 8-s − 2.24·9-s + 3.43·10-s + 11-s + 0.867·12-s + 0.412·13-s + 2.50·14-s − 2.97·15-s + 16-s + 4.12·17-s + 2.24·18-s − 3.33·19-s − 3.43·20-s − 2.17·21-s − 22-s + 7.84·23-s − 0.867·24-s + 6.77·25-s − 0.412·26-s − 4.55·27-s − 2.50·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.500·3-s + 0.5·4-s − 1.53·5-s − 0.354·6-s − 0.946·7-s − 0.353·8-s − 0.749·9-s + 1.08·10-s + 0.301·11-s + 0.250·12-s + 0.114·13-s + 0.668·14-s − 0.768·15-s + 0.250·16-s + 1.00·17-s + 0.529·18-s − 0.766·19-s − 0.767·20-s − 0.473·21-s − 0.213·22-s + 1.63·23-s − 0.177·24-s + 1.35·25-s − 0.0808·26-s − 0.875·27-s − 0.473·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 946 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 946 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7676346067\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7676346067\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 43 | \( 1 + T \) |
| good | 3 | \( 1 - 0.867T + 3T^{2} \) |
| 5 | \( 1 + 3.43T + 5T^{2} \) |
| 7 | \( 1 + 2.50T + 7T^{2} \) |
| 13 | \( 1 - 0.412T + 13T^{2} \) |
| 17 | \( 1 - 4.12T + 17T^{2} \) |
| 19 | \( 1 + 3.33T + 19T^{2} \) |
| 23 | \( 1 - 7.84T + 23T^{2} \) |
| 29 | \( 1 - 6.04T + 29T^{2} \) |
| 31 | \( 1 - 10.5T + 31T^{2} \) |
| 37 | \( 1 + 7.30T + 37T^{2} \) |
| 41 | \( 1 - 8.11T + 41T^{2} \) |
| 47 | \( 1 - 6.72T + 47T^{2} \) |
| 53 | \( 1 - 6.68T + 53T^{2} \) |
| 59 | \( 1 + 14.5T + 59T^{2} \) |
| 61 | \( 1 - 4.03T + 61T^{2} \) |
| 67 | \( 1 - 3.95T + 67T^{2} \) |
| 71 | \( 1 - 3.26T + 71T^{2} \) |
| 73 | \( 1 + 10.5T + 73T^{2} \) |
| 79 | \( 1 + 16.4T + 79T^{2} \) |
| 83 | \( 1 + 13.3T + 83T^{2} \) |
| 89 | \( 1 - 13.3T + 89T^{2} \) |
| 97 | \( 1 - 6.94T + 97T^{2} \) |
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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
−9.994691766741042820623242089078, −8.900069764859201905247062674239, −8.532435282932071544100394194531, −7.67399563746616443769532214482, −6.92316822070135896508446230411, −5.99601067197919768465732264106, −4.52681220730753181885403711073, −3.37020477491713701372270427827, −2.84494357467408440458887096638, −0.73088896235550742786706163917,
0.73088896235550742786706163917, 2.84494357467408440458887096638, 3.37020477491713701372270427827, 4.52681220730753181885403711073, 5.99601067197919768465732264106, 6.92316822070135896508446230411, 7.67399563746616443769532214482, 8.532435282932071544100394194531, 8.900069764859201905247062674239, 9.994691766741042820623242089078
