L(s) = 1 | − 0.633·2-s − 3-s − 1.59·4-s + 3.72·5-s + 0.633·6-s − 0.819·7-s + 2.28·8-s + 9-s − 2.36·10-s − 5.95·11-s + 1.59·12-s + 13-s + 0.519·14-s − 3.72·15-s + 1.75·16-s + 0.220·17-s − 0.633·18-s + 3.58·19-s − 5.95·20-s + 0.819·21-s + 3.77·22-s − 6.39·23-s − 2.28·24-s + 8.88·25-s − 0.633·26-s − 27-s + 1.31·28-s + ⋯ |
L(s) = 1 | − 0.448·2-s − 0.577·3-s − 0.799·4-s + 1.66·5-s + 0.258·6-s − 0.309·7-s + 0.806·8-s + 0.333·9-s − 0.746·10-s − 1.79·11-s + 0.461·12-s + 0.277·13-s + 0.138·14-s − 0.962·15-s + 0.437·16-s + 0.0535·17-s − 0.149·18-s + 0.822·19-s − 1.33·20-s + 0.178·21-s + 0.804·22-s − 1.33·23-s − 0.465·24-s + 1.77·25-s − 0.124·26-s − 0.192·27-s + 0.247·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 103 | \( 1 - T \) |
good | 2 | \( 1 + 0.633T + 2T^{2} \) |
| 5 | \( 1 - 3.72T + 5T^{2} \) |
| 7 | \( 1 + 0.819T + 7T^{2} \) |
| 11 | \( 1 + 5.95T + 11T^{2} \) |
| 17 | \( 1 - 0.220T + 17T^{2} \) |
| 19 | \( 1 - 3.58T + 19T^{2} \) |
| 23 | \( 1 + 6.39T + 23T^{2} \) |
| 29 | \( 1 - 2.80T + 29T^{2} \) |
| 31 | \( 1 - 2.92T + 31T^{2} \) |
| 37 | \( 1 + 1.86T + 37T^{2} \) |
| 41 | \( 1 - 2.27T + 41T^{2} \) |
| 43 | \( 1 + 4.99T + 43T^{2} \) |
| 47 | \( 1 - 0.499T + 47T^{2} \) |
| 53 | \( 1 + 6.32T + 53T^{2} \) |
| 59 | \( 1 - 0.203T + 59T^{2} \) |
| 61 | \( 1 - 4.63T + 61T^{2} \) |
| 67 | \( 1 + 12.1T + 67T^{2} \) |
| 71 | \( 1 - 8.37T + 71T^{2} \) |
| 73 | \( 1 + 11.4T + 73T^{2} \) |
| 79 | \( 1 - 2.25T + 79T^{2} \) |
| 83 | \( 1 + 2.79T + 83T^{2} \) |
| 89 | \( 1 + 2.90T + 89T^{2} \) |
| 97 | \( 1 - 8.31T + 97T^{2} \) |
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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
−8.130898799668236026803328308697, −7.48679714376746883189147906840, −6.39570732684304392210651552491, −5.78711954911960621271248546308, −5.19927060165818096492707387258, −4.62315747269119398380724852428, −3.28152657099303442123393118823, −2.27794241905070812043351340347, −1.28420775373071288184895748740, 0,
1.28420775373071288184895748740, 2.27794241905070812043351340347, 3.28152657099303442123393118823, 4.62315747269119398380724852428, 5.19927060165818096492707387258, 5.78711954911960621271248546308, 6.39570732684304392210651552491, 7.48679714376746883189147906840, 8.130898799668236026803328308697