| L(s) = 1 | − 2.38·2-s − 1.75·3-s + 3.70·4-s − 3.96·5-s + 4.18·6-s − 4.47·7-s − 4.06·8-s + 0.0656·9-s + 9.47·10-s + 1.22·11-s − 6.48·12-s + 5.68·13-s + 10.6·14-s + 6.94·15-s + 2.31·16-s + 0.875·17-s − 0.156·18-s + 0.266·19-s − 14.6·20-s + 7.82·21-s − 2.92·22-s + 4.72·23-s + 7.12·24-s + 10.7·25-s − 13.5·26-s + 5.13·27-s − 16.5·28-s + ⋯ |
| L(s) = 1 | − 1.68·2-s − 1.01·3-s + 1.85·4-s − 1.77·5-s + 1.70·6-s − 1.68·7-s − 1.43·8-s + 0.0218·9-s + 2.99·10-s + 0.369·11-s − 1.87·12-s + 1.57·13-s + 2.85·14-s + 1.79·15-s + 0.577·16-s + 0.212·17-s − 0.0369·18-s + 0.0611·19-s − 3.28·20-s + 1.70·21-s − 0.624·22-s + 0.985·23-s + 1.45·24-s + 2.14·25-s − 2.66·26-s + 0.988·27-s − 3.12·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5077 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5077 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2336704957\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2336704957\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5077 | \( 1+O(T) \) |
| good | 2 | \( 1 + 2.38T + 2T^{2} \) |
| 3 | \( 1 + 1.75T + 3T^{2} \) |
| 5 | \( 1 + 3.96T + 5T^{2} \) |
| 7 | \( 1 + 4.47T + 7T^{2} \) |
| 11 | \( 1 - 1.22T + 11T^{2} \) |
| 13 | \( 1 - 5.68T + 13T^{2} \) |
| 17 | \( 1 - 0.875T + 17T^{2} \) |
| 19 | \( 1 - 0.266T + 19T^{2} \) |
| 23 | \( 1 - 4.72T + 23T^{2} \) |
| 29 | \( 1 - 3.52T + 29T^{2} \) |
| 31 | \( 1 + 5.19T + 31T^{2} \) |
| 37 | \( 1 - 0.569T + 37T^{2} \) |
| 41 | \( 1 - 7.28T + 41T^{2} \) |
| 43 | \( 1 + 1.60T + 43T^{2} \) |
| 47 | \( 1 - 6.34T + 47T^{2} \) |
| 53 | \( 1 + 6.00T + 53T^{2} \) |
| 59 | \( 1 - 10.6T + 59T^{2} \) |
| 61 | \( 1 + 14.5T + 61T^{2} \) |
| 67 | \( 1 - 1.39T + 67T^{2} \) |
| 71 | \( 1 - 2.69T + 71T^{2} \) |
| 73 | \( 1 - 5.82T + 73T^{2} \) |
| 79 | \( 1 + 14.5T + 79T^{2} \) |
| 83 | \( 1 - 4.36T + 83T^{2} \) |
| 89 | \( 1 + 3.79T + 89T^{2} \) |
| 97 | \( 1 + 2.58T + 97T^{2} \) |
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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
−8.394556628288160467363297283971, −7.52178205221093027466128851053, −6.91824722429166118235980199896, −6.42014783168318355840920290793, −5.73297825554772219025196121904, −4.39288577052552244435984616102, −3.51746503296834125893801858982, −2.91376046539472852501123977380, −1.06363122683133242600383688631, −0.45576638146985676523572438859,
0.45576638146985676523572438859, 1.06363122683133242600383688631, 2.91376046539472852501123977380, 3.51746503296834125893801858982, 4.39288577052552244435984616102, 5.73297825554772219025196121904, 6.42014783168318355840920290793, 6.91824722429166118235980199896, 7.52178205221093027466128851053, 8.394556628288160467363297283971
