| L(s) = 1 | + 43.1·2-s + 195.·3-s + 1.34e3·4-s + 457.·5-s + 8.44e3·6-s − 6.91e3·7-s + 3.61e4·8-s + 1.86e4·9-s + 1.97e4·10-s + 8.57e4·11-s + 2.64e5·12-s − 2.98e5·14-s + 8.95e4·15-s + 8.68e5·16-s + 3.66e5·17-s + 8.03e5·18-s − 6.20e5·19-s + 6.17e5·20-s − 1.35e6·21-s + 3.69e6·22-s − 7.34e5·23-s + 7.07e6·24-s − 1.74e6·25-s − 2.08e5·27-s − 9.32e6·28-s − 1.77e6·29-s + 3.86e6·30-s + ⋯ |
| L(s) = 1 | + 1.90·2-s + 1.39·3-s + 2.63·4-s + 0.327·5-s + 2.66·6-s − 1.08·7-s + 3.12·8-s + 0.945·9-s + 0.624·10-s + 1.76·11-s + 3.67·12-s − 2.07·14-s + 0.456·15-s + 3.31·16-s + 1.06·17-s + 1.80·18-s − 1.09·19-s + 0.863·20-s − 1.51·21-s + 3.36·22-s − 0.547·23-s + 4.35·24-s − 0.892·25-s − 0.0754·27-s − 2.86·28-s − 0.465·29-s + 0.871·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(13.42431558\) |
| \(L(\frac12)\) |
\(\approx\) |
\(13.42431558\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 \) |
| good | 2 | \( 1 - 43.1T + 512T^{2} \) |
| 3 | \( 1 - 195.T + 1.96e4T^{2} \) |
| 5 | \( 1 - 457.T + 1.95e6T^{2} \) |
| 7 | \( 1 + 6.91e3T + 4.03e7T^{2} \) |
| 11 | \( 1 - 8.57e4T + 2.35e9T^{2} \) |
| 17 | \( 1 - 3.66e5T + 1.18e11T^{2} \) |
| 19 | \( 1 + 6.20e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 7.34e5T + 1.80e12T^{2} \) |
| 29 | \( 1 + 1.77e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 1.44e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 1.41e7T + 1.29e14T^{2} \) |
| 41 | \( 1 + 4.75e6T + 3.27e14T^{2} \) |
| 43 | \( 1 - 1.49e6T + 5.02e14T^{2} \) |
| 47 | \( 1 - 2.44e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 8.17e6T + 3.29e15T^{2} \) |
| 59 | \( 1 - 8.34e7T + 8.66e15T^{2} \) |
| 61 | \( 1 + 4.23e7T + 1.16e16T^{2} \) |
| 67 | \( 1 + 3.16e8T + 2.72e16T^{2} \) |
| 71 | \( 1 - 3.21e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 1.58e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 5.13e7T + 1.19e17T^{2} \) |
| 83 | \( 1 + 1.52e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 2.74e7T + 3.50e17T^{2} \) |
| 97 | \( 1 - 9.88e8T + 7.60e17T^{2} \) |
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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
−11.58186988600338413341996122358, −10.08987475114887998091616009904, −9.148110447920724160187991894977, −7.68532359744006531526567046565, −6.56032882680550978187844713594, −5.85982474616447159849407807279, −4.07978769937658459607947029046, −3.62904363312926441995394012327, −2.61270960969382257867799492446, −1.60919672769291290400601981386,
1.60919672769291290400601981386, 2.61270960969382257867799492446, 3.62904363312926441995394012327, 4.07978769937658459607947029046, 5.85982474616447159849407807279, 6.56032882680550978187844713594, 7.68532359744006531526567046565, 9.148110447920724160187991894977, 10.08987475114887998091616009904, 11.58186988600338413341996122358
