| L(s) = 1 | + 2.81·2-s + 5.91·4-s − 1.28·5-s + 11.0·8-s − 3.62·10-s − 4.20·11-s + 13-s + 19.1·16-s + 1.62·17-s + 6.33·19-s − 7.62·20-s − 11.8·22-s + 2.71·23-s − 3.33·25-s + 2.81·26-s + 4.33·29-s + 7.49·31-s + 31.9·32-s + 4.57·34-s + 3.42·37-s + 17.8·38-s − 14.2·40-s − 7.62·41-s − 4.91·43-s − 24.8·44-s + 7.62·46-s − 1.08·47-s + ⋯ |
| L(s) = 1 | + 1.98·2-s + 2.95·4-s − 0.576·5-s + 3.89·8-s − 1.14·10-s − 1.26·11-s + 0.277·13-s + 4.79·16-s + 0.394·17-s + 1.45·19-s − 1.70·20-s − 2.52·22-s + 0.565·23-s − 0.667·25-s + 0.551·26-s + 0.805·29-s + 1.34·31-s + 5.63·32-s + 0.785·34-s + 0.562·37-s + 2.89·38-s − 2.24·40-s − 1.19·41-s − 0.749·43-s − 3.75·44-s + 1.12·46-s − 0.158·47-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
| \(L(1)\) |
\(\approx\) |
\(7.597942751\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.597942751\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.81T + 2T^{2} \) |
| 5 | \( 1 + 1.28T + 5T^{2} \) |
| 11 | \( 1 + 4.20T + 11T^{2} \) |
| 17 | \( 1 - 1.62T + 17T^{2} \) |
| 19 | \( 1 - 6.33T + 19T^{2} \) |
| 23 | \( 1 - 2.71T + 23T^{2} \) |
| 29 | \( 1 - 4.33T + 29T^{2} \) |
| 31 | \( 1 - 7.49T + 31T^{2} \) |
| 37 | \( 1 - 3.42T + 37T^{2} \) |
| 41 | \( 1 + 7.62T + 41T^{2} \) |
| 43 | \( 1 + 4.91T + 43T^{2} \) |
| 47 | \( 1 + 1.08T + 47T^{2} \) |
| 53 | \( 1 + 1.75T + 53T^{2} \) |
| 59 | \( 1 + 4.57T + 59T^{2} \) |
| 61 | \( 1 - 5.25T + 61T^{2} \) |
| 67 | \( 1 - 8.67T + 67T^{2} \) |
| 71 | \( 1 - 6.78T + 71T^{2} \) |
| 73 | \( 1 + 4.07T + 73T^{2} \) |
| 79 | \( 1 + 8.91T + 79T^{2} \) |
| 83 | \( 1 + 4.33T + 83T^{2} \) |
| 89 | \( 1 - 4.54T + 89T^{2} \) |
| 97 | \( 1 + 11.3T + 97T^{2} \) |
| 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
−7.81000075767196021504876253078, −7.25446309905954758931932749486, −6.49128606903697864440430499550, −5.74877280181030637065915291831, −5.05194979327241916724434928685, −4.65453411340636548670104594883, −3.60850136308740095399475159724, −3.12408052587955698750533429961, −2.38895819171670406518577495978, −1.15803120692999356150980193738,
1.15803120692999356150980193738, 2.38895819171670406518577495978, 3.12408052587955698750533429961, 3.60850136308740095399475159724, 4.65453411340636548670104594883, 5.05194979327241916724434928685, 5.74877280181030637065915291831, 6.49128606903697864440430499550, 7.25446309905954758931932749486, 7.81000075767196021504876253078
