L(s) = 1 | − 2-s − 3-s − 4-s + 5-s + 6-s + 7-s + 3·8-s + 9-s − 10-s + 12-s + 5·13-s − 14-s − 15-s − 16-s − 7·17-s − 18-s − 6·19-s − 20-s − 21-s − 4·23-s − 3·24-s − 4·25-s − 5·26-s − 27-s − 28-s + 9·29-s + 30-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s − 1/2·4-s + 0.447·5-s + 0.408·6-s + 0.377·7-s + 1.06·8-s + 1/3·9-s − 0.316·10-s + 0.288·12-s + 1.38·13-s − 0.267·14-s − 0.258·15-s − 1/4·16-s − 1.69·17-s − 0.235·18-s − 1.37·19-s − 0.223·20-s − 0.218·21-s − 0.834·23-s − 0.612·24-s − 4/5·25-s − 0.980·26-s − 0.192·27-s − 0.188·28-s + 1.67·29-s + 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2541 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2541 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 + 7 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 9 T + p T^{2} \) |
| 41 | \( 1 + 7 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 - 2 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 9 T + p T^{2} \) |
| 97 | \( 1 - 17 T + p T^{2} \) |
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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
−8.335179887542289269197168685667, −8.281616083978827853312545546455, −6.84791460647269591102539522885, −6.32746718081635065128127364512, −5.45706459447650445840299759081, −4.41382093494792401463394752388, −4.02399247865742322166262781348, −2.26554886835817034792614924163, −1.33573165844543695115337732937, 0,
1.33573165844543695115337732937, 2.26554886835817034792614924163, 4.02399247865742322166262781348, 4.41382093494792401463394752388, 5.45706459447650445840299759081, 6.32746718081635065128127364512, 6.84791460647269591102539522885, 8.281616083978827853312545546455, 8.335179887542289269197168685667