| L(s) = 1 | + 2.60·2-s − 1.02·3-s + 4.78·4-s + 1.68·5-s − 2.65·6-s + 2.80·7-s + 7.25·8-s − 1.95·9-s + 4.37·10-s − 1.84·11-s − 4.88·12-s − 0.773·13-s + 7.31·14-s − 1.71·15-s + 9.33·16-s − 0.882·17-s − 5.10·18-s − 3.94·19-s + 8.04·20-s − 2.86·21-s − 4.81·22-s − 23-s − 7.40·24-s − 2.17·25-s − 2.01·26-s + 5.06·27-s + 13.4·28-s + ⋯ |
| L(s) = 1 | + 1.84·2-s − 0.589·3-s + 2.39·4-s + 0.751·5-s − 1.08·6-s + 1.06·7-s + 2.56·8-s − 0.652·9-s + 1.38·10-s − 0.557·11-s − 1.40·12-s − 0.214·13-s + 1.95·14-s − 0.442·15-s + 2.33·16-s − 0.214·17-s − 1.20·18-s − 0.904·19-s + 1.79·20-s − 0.625·21-s − 1.02·22-s − 0.208·23-s − 1.51·24-s − 0.435·25-s − 0.395·26-s + 0.973·27-s + 2.53·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 667 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 667 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.141599521\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.141599521\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 23 | \( 1 + T \) |
| 29 | \( 1 - T \) |
| good | 2 | \( 1 - 2.60T + 2T^{2} \) |
| 3 | \( 1 + 1.02T + 3T^{2} \) |
| 5 | \( 1 - 1.68T + 5T^{2} \) |
| 7 | \( 1 - 2.80T + 7T^{2} \) |
| 11 | \( 1 + 1.84T + 11T^{2} \) |
| 13 | \( 1 + 0.773T + 13T^{2} \) |
| 17 | \( 1 + 0.882T + 17T^{2} \) |
| 19 | \( 1 + 3.94T + 19T^{2} \) |
| 31 | \( 1 - 5.16T + 31T^{2} \) |
| 37 | \( 1 - 3.00T + 37T^{2} \) |
| 41 | \( 1 - 1.65T + 41T^{2} \) |
| 43 | \( 1 + 3.35T + 43T^{2} \) |
| 47 | \( 1 + 1.98T + 47T^{2} \) |
| 53 | \( 1 - 1.44T + 53T^{2} \) |
| 59 | \( 1 - 2.83T + 59T^{2} \) |
| 61 | \( 1 - 4.60T + 61T^{2} \) |
| 67 | \( 1 + 13.2T + 67T^{2} \) |
| 71 | \( 1 + 14.4T + 71T^{2} \) |
| 73 | \( 1 + 12.2T + 73T^{2} \) |
| 79 | \( 1 + 2.29T + 79T^{2} \) |
| 83 | \( 1 - 6.46T + 83T^{2} \) |
| 89 | \( 1 - 16.1T + 89T^{2} \) |
| 97 | \( 1 - 11.0T + 97T^{2} \) |
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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
−10.86990072869259468573743735986, −10.17788766549400742756681414591, −8.585539184287464522975735865945, −7.56515178881105083586507839354, −6.37393920337471017847370485241, −5.83037236769107164872535413769, −5.01616982020542523982133945178, −4.37283765396733770177848087478, −2.85949544639081381180237073440, −1.91658444091893646947894067103,
1.91658444091893646947894067103, 2.85949544639081381180237073440, 4.37283765396733770177848087478, 5.01616982020542523982133945178, 5.83037236769107164872535413769, 6.37393920337471017847370485241, 7.56515178881105083586507839354, 8.585539184287464522975735865945, 10.17788766549400742756681414591, 10.86990072869259468573743735986
