L(s) = 1 | + 2-s − 3-s + 4-s − 0.194·5-s − 6-s + 1.95·7-s + 8-s + 9-s − 0.194·10-s + 6.30·11-s − 12-s + 4.81·13-s + 1.95·14-s + 0.194·15-s + 16-s − 3.20·17-s + 18-s + 0.859·19-s − 0.194·20-s − 1.95·21-s + 6.30·22-s − 24-s − 4.96·25-s + 4.81·26-s − 27-s + 1.95·28-s + 5.41·29-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.0868·5-s − 0.408·6-s + 0.737·7-s + 0.353·8-s + 0.333·9-s − 0.0614·10-s + 1.90·11-s − 0.288·12-s + 1.33·13-s + 0.521·14-s + 0.0501·15-s + 0.250·16-s − 0.778·17-s + 0.235·18-s + 0.197·19-s − 0.0434·20-s − 0.425·21-s + 1.34·22-s − 0.204·24-s − 0.992·25-s + 0.943·26-s − 0.192·27-s + 0.368·28-s + 1.00·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3174 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3174 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.186787665\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.186787665\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 23 | \( 1 \) |
good | 5 | \( 1 + 0.194T + 5T^{2} \) |
| 7 | \( 1 - 1.95T + 7T^{2} \) |
| 11 | \( 1 - 6.30T + 11T^{2} \) |
| 13 | \( 1 - 4.81T + 13T^{2} \) |
| 17 | \( 1 + 3.20T + 17T^{2} \) |
| 19 | \( 1 - 0.859T + 19T^{2} \) |
| 29 | \( 1 - 5.41T + 29T^{2} \) |
| 31 | \( 1 + 1.01T + 31T^{2} \) |
| 37 | \( 1 - 6.89T + 37T^{2} \) |
| 41 | \( 1 + 0.972T + 41T^{2} \) |
| 43 | \( 1 + 5.14T + 43T^{2} \) |
| 47 | \( 1 + 7.41T + 47T^{2} \) |
| 53 | \( 1 - 2.00T + 53T^{2} \) |
| 59 | \( 1 + 8.60T + 59T^{2} \) |
| 61 | \( 1 - 12.1T + 61T^{2} \) |
| 67 | \( 1 - 7.09T + 67T^{2} \) |
| 71 | \( 1 - 1.10T + 71T^{2} \) |
| 73 | \( 1 + 13.1T + 73T^{2} \) |
| 79 | \( 1 - 1.12T + 79T^{2} \) |
| 83 | \( 1 - 4.00T + 83T^{2} \) |
| 89 | \( 1 + 9.55T + 89T^{2} \) |
| 97 | \( 1 + 1.12T + 97T^{2} \) |
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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
−8.602299094078072709854755219479, −7.904315204975310582739806058501, −6.77796755541019867278441694020, −6.41322948265320094130032900509, −5.70450439554827416314188231628, −4.66968881442184438520532032328, −4.12777972212021034289981633053, −3.38319761757124889788430098909, −1.88374512670597499121638051623, −1.11773203611966143124392243727,
1.11773203611966143124392243727, 1.88374512670597499121638051623, 3.38319761757124889788430098909, 4.12777972212021034289981633053, 4.66968881442184438520532032328, 5.70450439554827416314188231628, 6.41322948265320094130032900509, 6.77796755541019867278441694020, 7.904315204975310582739806058501, 8.602299094078072709854755219479