| L(s) = 1 | − 42.7·2-s + 260.·3-s + 1.31e3·4-s − 1.11e4·6-s + 2.40e3·7-s − 3.44e4·8-s + 4.80e4·9-s + 4.15e4·11-s + 3.43e5·12-s − 1.03e5·13-s − 1.02e5·14-s + 8.00e5·16-s − 3.55e5·17-s − 2.05e6·18-s − 3.49e5·19-s + 6.24e5·21-s − 1.77e6·22-s − 2.28e5·23-s − 8.97e6·24-s + 4.44e6·26-s + 7.37e6·27-s + 3.16e6·28-s + 4.02e6·29-s − 3.29e6·31-s − 1.65e7·32-s + 1.08e7·33-s + 1.51e7·34-s + ⋯ |
| L(s) = 1 | − 1.89·2-s + 1.85·3-s + 2.57·4-s − 3.50·6-s + 0.377·7-s − 2.97·8-s + 2.44·9-s + 0.856·11-s + 4.77·12-s − 1.00·13-s − 0.714·14-s + 3.05·16-s − 1.03·17-s − 4.61·18-s − 0.614·19-s + 0.701·21-s − 1.61·22-s − 0.170·23-s − 5.52·24-s + 1.90·26-s + 2.67·27-s + 0.973·28-s + 1.05·29-s − 0.640·31-s − 2.79·32-s + 1.58·33-s + 1.95·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(2.078109965\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.078109965\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 7 | \( 1 - 2.40e3T \) |
| good | 2 | \( 1 + 42.7T + 512T^{2} \) |
| 3 | \( 1 - 260.T + 1.96e4T^{2} \) |
| 11 | \( 1 - 4.15e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + 1.03e5T + 1.06e10T^{2} \) |
| 17 | \( 1 + 3.55e5T + 1.18e11T^{2} \) |
| 19 | \( 1 + 3.49e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 2.28e5T + 1.80e12T^{2} \) |
| 29 | \( 1 - 4.02e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 3.29e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 2.13e7T + 1.29e14T^{2} \) |
| 41 | \( 1 - 1.05e7T + 3.27e14T^{2} \) |
| 43 | \( 1 + 3.89e6T + 5.02e14T^{2} \) |
| 47 | \( 1 - 3.31e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 6.31e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 5.86e7T + 8.66e15T^{2} \) |
| 61 | \( 1 - 1.37e8T + 1.16e16T^{2} \) |
| 67 | \( 1 + 8.36e7T + 2.72e16T^{2} \) |
| 71 | \( 1 - 1.50e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 2.74e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 2.75e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 2.32e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 2.71e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 9.76e8T + 7.60e17T^{2} \) |
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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
−10.48830115276007700894390499622, −9.595185387414605495457190946537, −8.975632593092579377835681868588, −8.279155876419919711337564842210, −7.44521521024186016906653226964, −6.63405887022888935006164635060, −4.14246296924503175458369494309, −2.60070010355269070872273085108, −2.06264267092577066350874277184, −0.873431912594651354936750251317,
0.873431912594651354936750251317, 2.06264267092577066350874277184, 2.60070010355269070872273085108, 4.14246296924503175458369494309, 6.63405887022888935006164635060, 7.44521521024186016906653226964, 8.279155876419919711337564842210, 8.975632593092579377835681868588, 9.595185387414605495457190946537, 10.48830115276007700894390499622
