L(s) = 1 | + 3.21·2-s + 3·3-s + 2.35·4-s + 9.65·6-s − 7·7-s − 18.1·8-s + 9·9-s − 2.09·11-s + 7.05·12-s + 80.8·13-s − 22.5·14-s − 77.2·16-s + 101.·17-s + 28.9·18-s + 143.·19-s − 21·21-s − 6.75·22-s − 116.·23-s − 54.5·24-s + 260.·26-s + 27·27-s − 16.4·28-s + 181.·29-s + 303.·31-s − 103.·32-s − 6.29·33-s + 328.·34-s + ⋯ |
L(s) = 1 | + 1.13·2-s + 0.577·3-s + 0.293·4-s + 0.656·6-s − 0.377·7-s − 0.803·8-s + 0.333·9-s − 0.0575·11-s + 0.169·12-s + 1.72·13-s − 0.429·14-s − 1.20·16-s + 1.45·17-s + 0.379·18-s + 1.73·19-s − 0.218·21-s − 0.0654·22-s − 1.05·23-s − 0.463·24-s + 1.96·26-s + 0.192·27-s − 0.111·28-s + 1.16·29-s + 1.75·31-s − 0.570·32-s − 0.0332·33-s + 1.65·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(4.268879064\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.268879064\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - 3T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + 7T \) |
good | 2 | \( 1 - 3.21T + 8T^{2} \) |
| 11 | \( 1 + 2.09T + 1.33e3T^{2} \) |
| 13 | \( 1 - 80.8T + 2.19e3T^{2} \) |
| 17 | \( 1 - 101.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 143.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 116.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 181.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 303.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 158.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 379.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 238.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 125.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 43.4T + 1.48e5T^{2} \) |
| 59 | \( 1 - 31.0T + 2.05e5T^{2} \) |
| 61 | \( 1 + 812.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 426.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 1.03e3T + 3.57e5T^{2} \) |
| 73 | \( 1 + 471.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.20e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.32e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 886.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 134.T + 9.12e5T^{2} \) |
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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
−10.36280646987528883018884648694, −9.619099751523064496191199647871, −8.595661191527228201445873734401, −7.80564510107700429414145394121, −6.44088279490559081812844451776, −5.76402096855471449556149310098, −4.64225306097805184154814645734, −3.50238448430169783425923103055, −3.03619470058614862719033602550, −1.14647755478358344269672677464,
1.14647755478358344269672677464, 3.03619470058614862719033602550, 3.50238448430169783425923103055, 4.64225306097805184154814645734, 5.76402096855471449556149310098, 6.44088279490559081812844451776, 7.80564510107700429414145394121, 8.595661191527228201445873734401, 9.619099751523064496191199647871, 10.36280646987528883018884648694