| L(s) = 1 | − 269.·2-s + 2.18e3·3-s + 3.97e4·4-s − 5.88e5·6-s − 2.21e6·7-s − 1.87e6·8-s + 4.78e6·9-s − 1.04e8·11-s + 8.69e7·12-s − 1.47e8·13-s + 5.97e8·14-s − 7.96e8·16-s − 1.95e9·17-s − 1.28e9·18-s − 6.19e9·19-s − 4.85e9·21-s + 2.81e10·22-s − 1.23e10·23-s − 4.10e9·24-s + 3.98e10·26-s + 1.04e10·27-s − 8.81e10·28-s + 1.52e10·29-s − 1.36e11·31-s + 2.76e11·32-s − 2.28e11·33-s + 5.26e11·34-s + ⋯ |
| L(s) = 1 | − 1.48·2-s + 0.577·3-s + 1.21·4-s − 0.858·6-s − 1.01·7-s − 0.316·8-s + 0.333·9-s − 1.61·11-s + 0.700·12-s − 0.654·13-s + 1.51·14-s − 0.742·16-s − 1.15·17-s − 0.495·18-s − 1.58·19-s − 0.588·21-s + 2.40·22-s − 0.758·23-s − 0.182·24-s + 0.972·26-s + 0.192·27-s − 1.23·28-s + 0.163·29-s − 0.894·31-s + 1.42·32-s − 0.933·33-s + 1.71·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(16-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+15/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(8)\) |
\(\approx\) |
\(0.08134775002\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.08134775002\) |
| \(L(\frac{17}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 2.18e3T \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + 269.T + 3.27e4T^{2} \) |
| 7 | \( 1 + 2.21e6T + 4.74e12T^{2} \) |
| 11 | \( 1 + 1.04e8T + 4.17e15T^{2} \) |
| 13 | \( 1 + 1.47e8T + 5.11e16T^{2} \) |
| 17 | \( 1 + 1.95e9T + 2.86e18T^{2} \) |
| 19 | \( 1 + 6.19e9T + 1.51e19T^{2} \) |
| 23 | \( 1 + 1.23e10T + 2.66e20T^{2} \) |
| 29 | \( 1 - 1.52e10T + 8.62e21T^{2} \) |
| 31 | \( 1 + 1.36e11T + 2.34e22T^{2} \) |
| 37 | \( 1 - 7.49e11T + 3.33e23T^{2} \) |
| 41 | \( 1 - 2.69e11T + 1.55e24T^{2} \) |
| 43 | \( 1 - 5.96e11T + 3.17e24T^{2} \) |
| 47 | \( 1 + 4.90e12T + 1.20e25T^{2} \) |
| 53 | \( 1 - 2.94e12T + 7.31e25T^{2} \) |
| 59 | \( 1 + 9.17e12T + 3.65e26T^{2} \) |
| 61 | \( 1 + 3.22e13T + 6.02e26T^{2} \) |
| 67 | \( 1 + 3.17e13T + 2.46e27T^{2} \) |
| 71 | \( 1 + 1.02e14T + 5.87e27T^{2} \) |
| 73 | \( 1 - 8.48e13T + 8.90e27T^{2} \) |
| 79 | \( 1 + 8.38e13T + 2.91e28T^{2} \) |
| 83 | \( 1 - 3.37e14T + 6.11e28T^{2} \) |
| 89 | \( 1 + 7.98e14T + 1.74e29T^{2} \) |
| 97 | \( 1 - 6.39e14T + 6.33e29T^{2} \) |
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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
−10.94137758389000126377948762619, −10.16473039838464423802987368228, −9.315313057590068430744058276424, −8.331324466898040834713361089434, −7.46404650924128806359468266707, −6.32913043478487185605863257041, −4.47476980088687212046698674693, −2.74848325322856210270727747787, −1.98940288391584193762345487046, −0.15319200130417824009128243223,
0.15319200130417824009128243223, 1.98940288391584193762345487046, 2.74848325322856210270727747787, 4.47476980088687212046698674693, 6.32913043478487185605863257041, 7.46404650924128806359468266707, 8.331324466898040834713361089434, 9.315313057590068430744058276424, 10.16473039838464423802987368228, 10.94137758389000126377948762619
