| L(s) = 1 | − 2.63·2-s + 4.96·4-s + 3.29·7-s − 7.82·8-s − 5.06·11-s − 1.39·13-s − 8.68·14-s + 10.7·16-s + 6.93·17-s − 3.60·19-s + 13.3·22-s + 23-s + 3.67·26-s + 16.3·28-s − 2.05·29-s − 3.78·31-s − 12.6·32-s − 18.3·34-s + 11.2·37-s + 9.51·38-s − 1.42·41-s − 7.57·43-s − 25.1·44-s − 2.63·46-s − 0.937·47-s + 3.83·49-s − 6.90·52-s + ⋯ |
| L(s) = 1 | − 1.86·2-s + 2.48·4-s + 1.24·7-s − 2.76·8-s − 1.52·11-s − 0.385·13-s − 2.32·14-s + 2.68·16-s + 1.68·17-s − 0.826·19-s + 2.84·22-s + 0.208·23-s + 0.719·26-s + 3.08·28-s − 0.382·29-s − 0.680·31-s − 2.23·32-s − 3.14·34-s + 1.84·37-s + 1.54·38-s − 0.222·41-s − 1.15·43-s − 3.78·44-s − 0.389·46-s − 0.136·47-s + 0.547·49-s − 0.957·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7538557873\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7538557873\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 2 | \( 1 + 2.63T + 2T^{2} \) |
| 7 | \( 1 - 3.29T + 7T^{2} \) |
| 11 | \( 1 + 5.06T + 11T^{2} \) |
| 13 | \( 1 + 1.39T + 13T^{2} \) |
| 17 | \( 1 - 6.93T + 17T^{2} \) |
| 19 | \( 1 + 3.60T + 19T^{2} \) |
| 29 | \( 1 + 2.05T + 29T^{2} \) |
| 31 | \( 1 + 3.78T + 31T^{2} \) |
| 37 | \( 1 - 11.2T + 37T^{2} \) |
| 41 | \( 1 + 1.42T + 41T^{2} \) |
| 43 | \( 1 + 7.57T + 43T^{2} \) |
| 47 | \( 1 + 0.937T + 47T^{2} \) |
| 53 | \( 1 + 0.143T + 53T^{2} \) |
| 59 | \( 1 - 3.20T + 59T^{2} \) |
| 61 | \( 1 - 5.22T + 61T^{2} \) |
| 67 | \( 1 + 14.7T + 67T^{2} \) |
| 71 | \( 1 + 3.57T + 71T^{2} \) |
| 73 | \( 1 + 0.0659T + 73T^{2} \) |
| 79 | \( 1 - 16.3T + 79T^{2} \) |
| 83 | \( 1 + 0.439T + 83T^{2} \) |
| 89 | \( 1 - 8.21T + 89T^{2} \) |
| 97 | \( 1 - 9.42T + 97T^{2} \) |
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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
−8.112076770398107905898928161091, −7.72884973260377371141164882884, −7.35820406567662238869430474819, −6.24487722548974570215745593970, −5.47104534662245550967511810542, −4.74185588073650037970721180861, −3.28995321715952124922929882832, −2.38344957877817988169252468426, −1.69201810164252663486875508493, −0.62605450261852490887545447240,
0.62605450261852490887545447240, 1.69201810164252663486875508493, 2.38344957877817988169252468426, 3.28995321715952124922929882832, 4.74185588073650037970721180861, 5.47104534662245550967511810542, 6.24487722548974570215745593970, 7.35820406567662238869430474819, 7.72884973260377371141164882884, 8.112076770398107905898928161091
