| L(s) = 1 | − 2-s + 3-s + 4-s − 5-s − 6-s − 8-s + 9-s + 10-s − 3.65·11-s + 12-s + 0.851·13-s − 15-s + 16-s + 6.16·17-s − 18-s − 5.50·19-s − 20-s + 3.65·22-s − 23-s − 24-s − 4·25-s − 0.851·26-s + 27-s + 4.14·29-s + 30-s − 3.35·31-s − 32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.447·5-s − 0.408·6-s − 0.353·8-s + 0.333·9-s + 0.316·10-s − 1.10·11-s + 0.288·12-s + 0.236·13-s − 0.258·15-s + 0.250·16-s + 1.49·17-s − 0.235·18-s − 1.26·19-s − 0.223·20-s + 0.779·22-s − 0.208·23-s − 0.204·24-s − 0.800·25-s − 0.167·26-s + 0.192·27-s + 0.770·29-s + 0.182·30-s − 0.603·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6762 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6762 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 5 | \( 1 + T + 5T^{2} \) |
| 11 | \( 1 + 3.65T + 11T^{2} \) |
| 13 | \( 1 - 0.851T + 13T^{2} \) |
| 17 | \( 1 - 6.16T + 17T^{2} \) |
| 19 | \( 1 + 5.50T + 19T^{2} \) |
| 29 | \( 1 - 4.14T + 29T^{2} \) |
| 31 | \( 1 + 3.35T + 31T^{2} \) |
| 37 | \( 1 - 6.81T + 37T^{2} \) |
| 41 | \( 1 - 2.49T + 41T^{2} \) |
| 43 | \( 1 - 7.30T + 43T^{2} \) |
| 47 | \( 1 + 2.35T + 47T^{2} \) |
| 53 | \( 1 - 6.30T + 53T^{2} \) |
| 59 | \( 1 + 3.16T + 59T^{2} \) |
| 61 | \( 1 + 10T + 61T^{2} \) |
| 67 | \( 1 + 10.3T + 67T^{2} \) |
| 71 | \( 1 + 8.45T + 71T^{2} \) |
| 73 | \( 1 + 1.34T + 73T^{2} \) |
| 79 | \( 1 + 11.4T + 79T^{2} \) |
| 83 | \( 1 - 7.65T + 83T^{2} \) |
| 89 | \( 1 - 6.81T + 89T^{2} \) |
| 97 | \( 1 + 0.444T + 97T^{2} \) |
| 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
−7.68567295364639754076261775126, −7.39180015764187245219969529913, −6.22401075285146325147018363873, −5.71884888552695080086015297105, −4.64147146246157067857948223365, −3.86663962663737701080918623063, −2.98837117641336030369224435805, −2.32172878093346651093307815324, −1.24479070695563246709164466722, 0,
1.24479070695563246709164466722, 2.32172878093346651093307815324, 2.98837117641336030369224435805, 3.86663962663737701080918623063, 4.64147146246157067857948223365, 5.71884888552695080086015297105, 6.22401075285146325147018363873, 7.39180015764187245219969529913, 7.68567295364639754076261775126
