L(s) = 1 | + 0.347·2-s + 1.53·3-s − 0.879·4-s + 5-s + 0.532·6-s − 1.87·7-s − 0.652·8-s + 1.34·9-s + 0.347·10-s − 1.34·12-s − 13-s − 0.652·14-s + 1.53·15-s + 0.652·16-s + 0.467·18-s − 19-s − 0.879·20-s − 2.87·21-s − 24-s + 25-s − 0.347·26-s + 0.532·27-s + 1.65·28-s + 1.53·29-s + 0.532·30-s + 0.879·32-s − 1.87·35-s + ⋯ |
L(s) = 1 | + 0.347·2-s + 1.53·3-s − 0.879·4-s + 5-s + 0.532·6-s − 1.87·7-s − 0.652·8-s + 1.34·9-s + 0.347·10-s − 1.34·12-s − 13-s − 0.652·14-s + 1.53·15-s + 0.652·16-s + 0.467·18-s − 19-s − 0.879·20-s − 2.87·21-s − 24-s + 25-s − 0.347·26-s + 0.532·27-s + 1.65·28-s + 1.53·29-s + 0.532·30-s + 0.879·32-s − 1.87·35-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
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.150640405\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.150640405\) |
\(L(1)\) |
|
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 | \( 1 - 0.347T + T^{2} \) |
| 3 | \( 1 - 1.53T + T^{2} \) |
| 7 | \( 1 + 1.87T + T^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 13 | \( 1 + T + T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 + T + T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 - 1.53T + T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - 0.347T + T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 + 1.87T + T^{2} \) |
| 59 | \( 1 - 0.347T + T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 71 | \( 1 + T + T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + 1.87T + T^{2} \) |
| 97 | \( 1 - 0.347T + T^{2} \) |
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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
−12.54844282431596209625602119164, −10.22902589876060478282363114307, −9.726120685350341513084077659429, −9.149549491802112850629565265460, −8.352929164953226007503362542103, −6.92443840811092655576461129943, −5.98012143876439665013731833059, −4.52874915155761929951101455354, −3.29338205648354503099323886350, −2.54225804609270401666256445757,
2.54225804609270401666256445757, 3.29338205648354503099323886350, 4.52874915155761929951101455354, 5.98012143876439665013731833059, 6.92443840811092655576461129943, 8.352929164953226007503362542103, 9.149549491802112850629565265460, 9.726120685350341513084077659429, 10.22902589876060478282363114307, 12.54844282431596209625602119164