| L(s) = 1 | − 1.86·2-s + 1.48·4-s − 0.866·5-s + 0.965·8-s + 1.61·10-s + 3.86·11-s + 13-s − 4.76·16-s + 3.34·17-s − 5.38·19-s − 1.28·20-s − 7.21·22-s + 5.24·23-s − 4.24·25-s − 1.86·26-s − 1.69·29-s + 7.56·31-s + 6.96·32-s − 6.24·34-s − 4.83·37-s + 10.0·38-s − 0.836·40-s + 4.06·41-s + 4.03·43-s + 5.73·44-s − 9.79·46-s + 3.65·47-s + ⋯ |
| L(s) = 1 | − 1.31·2-s + 0.741·4-s − 0.387·5-s + 0.341·8-s + 0.511·10-s + 1.16·11-s + 0.277·13-s − 1.19·16-s + 0.812·17-s − 1.23·19-s − 0.287·20-s − 1.53·22-s + 1.09·23-s − 0.849·25-s − 0.365·26-s − 0.315·29-s + 1.35·31-s + 1.23·32-s − 1.07·34-s − 0.794·37-s + 1.62·38-s − 0.132·40-s + 0.635·41-s + 0.615·43-s + 0.864·44-s − 1.44·46-s + 0.532·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5733 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5733 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9242358824\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9242358824\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 2 | \( 1 + 1.86T + 2T^{2} \) |
| 5 | \( 1 + 0.866T + 5T^{2} \) |
| 11 | \( 1 - 3.86T + 11T^{2} \) |
| 17 | \( 1 - 3.34T + 17T^{2} \) |
| 19 | \( 1 + 5.38T + 19T^{2} \) |
| 23 | \( 1 - 5.24T + 23T^{2} \) |
| 29 | \( 1 + 1.69T + 29T^{2} \) |
| 31 | \( 1 - 7.56T + 31T^{2} \) |
| 37 | \( 1 + 4.83T + 37T^{2} \) |
| 41 | \( 1 - 4.06T + 41T^{2} \) |
| 43 | \( 1 - 4.03T + 43T^{2} \) |
| 47 | \( 1 - 3.65T + 47T^{2} \) |
| 53 | \( 1 - 0.215T + 53T^{2} \) |
| 59 | \( 1 - 2.78T + 59T^{2} \) |
| 61 | \( 1 - 9.03T + 61T^{2} \) |
| 67 | \( 1 + 7.66T + 67T^{2} \) |
| 71 | \( 1 + 4.90T + 71T^{2} \) |
| 73 | \( 1 - 15.5T + 73T^{2} \) |
| 79 | \( 1 - 9.43T + 79T^{2} \) |
| 83 | \( 1 - 4.09T + 83T^{2} \) |
| 89 | \( 1 - 0.418T + 89T^{2} \) |
| 97 | \( 1 + 7.11T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.185299112261519812888806681689, −7.66493306979690661446249702432, −6.83039827726350170659677245459, −6.34982346256762812598143066620, −5.29052946426776852146781333068, −4.28217665730098311353027508560, −3.75039767693183788080030289587, −2.52515220688290210097092145447, −1.49187010768804836712566419059, −0.67820877841855453859683814695,
0.67820877841855453859683814695, 1.49187010768804836712566419059, 2.52515220688290210097092145447, 3.75039767693183788080030289587, 4.28217665730098311353027508560, 5.29052946426776852146781333068, 6.34982346256762812598143066620, 6.83039827726350170659677245459, 7.66493306979690661446249702432, 8.185299112261519812888806681689
