| L(s) = 1 | + 2-s − 3-s + 4-s + 5-s − 6-s + 3.87·7-s + 8-s + 9-s + 10-s − 1.53·11-s − 12-s + 4.10·13-s + 3.87·14-s − 15-s + 16-s + 18-s + 19-s + 20-s − 3.87·21-s − 1.53·22-s + 9.41·23-s − 24-s + 25-s + 4.10·26-s − 27-s + 3.87·28-s + 5.75·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.447·5-s − 0.408·6-s + 1.46·7-s + 0.353·8-s + 0.333·9-s + 0.316·10-s − 0.461·11-s − 0.288·12-s + 1.13·13-s + 1.03·14-s − 0.258·15-s + 0.250·16-s + 0.235·18-s + 0.229·19-s + 0.223·20-s − 0.846·21-s − 0.326·22-s + 1.96·23-s − 0.204·24-s + 0.200·25-s + 0.805·26-s − 0.192·27-s + 0.733·28-s + 1.06·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8670 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8670 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.296860062\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.296860062\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 17 | \( 1 \) |
| good | 7 | \( 1 - 3.87T + 7T^{2} \) |
| 11 | \( 1 + 1.53T + 11T^{2} \) |
| 13 | \( 1 - 4.10T + 13T^{2} \) |
| 19 | \( 1 - T + 19T^{2} \) |
| 23 | \( 1 - 9.41T + 23T^{2} \) |
| 29 | \( 1 - 5.75T + 29T^{2} \) |
| 31 | \( 1 - 6.45T + 31T^{2} \) |
| 37 | \( 1 - 0.305T + 37T^{2} \) |
| 41 | \( 1 + 11.5T + 41T^{2} \) |
| 43 | \( 1 + 12.5T + 43T^{2} \) |
| 47 | \( 1 - 5.63T + 47T^{2} \) |
| 53 | \( 1 - 4.22T + 53T^{2} \) |
| 59 | \( 1 + 13.0T + 59T^{2} \) |
| 61 | \( 1 - 4.61T + 61T^{2} \) |
| 67 | \( 1 + 4.08T + 67T^{2} \) |
| 71 | \( 1 - 14.2T + 71T^{2} \) |
| 73 | \( 1 + 15.4T + 73T^{2} \) |
| 79 | \( 1 + 7.19T + 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 - 6.24T + 89T^{2} \) |
| 97 | \( 1 - 2.24T + 97T^{2} \) |
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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
−7.66619929621399645843723229722, −6.87305879024654196258003229266, −6.33356897103346087335301492531, −5.52178164350560889795695876675, −4.89247729201507436518551691720, −4.67051290316123784273890518244, −3.51233919814767988646468598486, −2.72232242315711359393925438530, −1.62575282058971540297589664362, −1.05646576113070013183877034539,
1.05646576113070013183877034539, 1.62575282058971540297589664362, 2.72232242315711359393925438530, 3.51233919814767988646468598486, 4.67051290316123784273890518244, 4.89247729201507436518551691720, 5.52178164350560889795695876675, 6.33356897103346087335301492531, 6.87305879024654196258003229266, 7.66619929621399645843723229722
