L(s) = 1 | + 1.53·2-s − 1.87·3-s + 1.34·4-s + 5-s − 2.87·6-s + 0.347·7-s + 0.532·8-s + 2.53·9-s + 1.53·10-s − 2.53·12-s − 13-s + 0.532·14-s − 1.87·15-s − 0.532·16-s + 3.87·18-s − 19-s + 1.34·20-s − 0.652·21-s − 24-s + 25-s − 1.53·26-s − 2.87·27-s + 0.467·28-s − 1.87·29-s − 2.87·30-s − 1.34·32-s + 0.347·35-s + ⋯ |
L(s) = 1 | + 1.53·2-s − 1.87·3-s + 1.34·4-s + 5-s − 2.87·6-s + 0.347·7-s + 0.532·8-s + 2.53·9-s + 1.53·10-s − 2.53·12-s − 13-s + 0.532·14-s − 1.87·15-s − 0.532·16-s + 3.87·18-s − 19-s + 1.34·20-s − 0.652·21-s − 24-s + 25-s − 1.53·26-s − 2.87·27-s + 0.467·28-s − 1.87·29-s − 2.87·30-s − 1.34·32-s + 0.347·35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 335 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 335 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.135178248\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.135178248\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 - T \) |
| 67 | \( 1 - T \) |
good | 2 | \( 1 - 1.53T + T^{2} \) |
| 3 | \( 1 + 1.87T + T^{2} \) |
| 7 | \( 1 - 0.347T + T^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 13 | \( 1 + T + T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 + T + T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 + 1.87T + T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - 1.53T + T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 - 0.347T + T^{2} \) |
| 59 | \( 1 - 1.53T + T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 71 | \( 1 + T + T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 - 0.347T + T^{2} \) |
| 97 | \( 1 - 1.53T + T^{2} \) |
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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
−11.91356519612076555736568189707, −11.17263940596115141046726397149, −10.41798551668896672858128052855, −9.374343001607053144156478783777, −7.27354473602909558849204002413, −6.41843497820471896040389423874, −5.64131577602335512212995357024, −5.06977199406534976849675565564, −4.15281317513195459619515169215, −2.10045216621666588780373750316,
2.10045216621666588780373750316, 4.15281317513195459619515169215, 5.06977199406534976849675565564, 5.64131577602335512212995357024, 6.41843497820471896040389423874, 7.27354473602909558849204002413, 9.374343001607053144156478783777, 10.41798551668896672858128052855, 11.17263940596115141046726397149, 11.91356519612076555736568189707