| L(s) = 1 | + 1.06·2-s − 3.37·3-s − 0.862·4-s − 3.60·6-s + 2.91·7-s − 3.05·8-s + 8.41·9-s + 2.52·11-s + 2.91·12-s + 0.109·13-s + 3.10·14-s − 1.52·16-s − 6.38·17-s + 8.97·18-s − 6.56·19-s − 9.85·21-s + 2.69·22-s − 3.08·23-s + 10.3·24-s + 0.116·26-s − 18.3·27-s − 2.51·28-s + 29-s + 1.18·31-s + 4.47·32-s − 8.52·33-s − 6.80·34-s + ⋯ |
| L(s) = 1 | + 0.754·2-s − 1.95·3-s − 0.431·4-s − 1.47·6-s + 1.10·7-s − 1.07·8-s + 2.80·9-s + 0.760·11-s + 0.841·12-s + 0.0302·13-s + 0.830·14-s − 0.382·16-s − 1.54·17-s + 2.11·18-s − 1.50·19-s − 2.14·21-s + 0.573·22-s − 0.643·23-s + 2.10·24-s + 0.0228·26-s − 3.52·27-s − 0.475·28-s + 0.185·29-s + 0.213·31-s + 0.791·32-s − 1.48·33-s − 1.16·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 725 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 725 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 29 | \( 1 - T \) |
| good | 2 | \( 1 - 1.06T + 2T^{2} \) |
| 3 | \( 1 + 3.37T + 3T^{2} \) |
| 7 | \( 1 - 2.91T + 7T^{2} \) |
| 11 | \( 1 - 2.52T + 11T^{2} \) |
| 13 | \( 1 - 0.109T + 13T^{2} \) |
| 17 | \( 1 + 6.38T + 17T^{2} \) |
| 19 | \( 1 + 6.56T + 19T^{2} \) |
| 23 | \( 1 + 3.08T + 23T^{2} \) |
| 31 | \( 1 - 1.18T + 31T^{2} \) |
| 37 | \( 1 + 4.65T + 37T^{2} \) |
| 41 | \( 1 - 2.26T + 41T^{2} \) |
| 43 | \( 1 + 10.1T + 43T^{2} \) |
| 47 | \( 1 + 4.78T + 47T^{2} \) |
| 53 | \( 1 - 10.3T + 53T^{2} \) |
| 59 | \( 1 + 7.38T + 59T^{2} \) |
| 61 | \( 1 + 10.2T + 61T^{2} \) |
| 67 | \( 1 + 4.55T + 67T^{2} \) |
| 71 | \( 1 + 1.42T + 71T^{2} \) |
| 73 | \( 1 - 3.96T + 73T^{2} \) |
| 79 | \( 1 + 1.47T + 79T^{2} \) |
| 83 | \( 1 - 2.72T + 83T^{2} \) |
| 89 | \( 1 + 0.228T + 89T^{2} \) |
| 97 | \( 1 - 8.49T + 97T^{2} \) |
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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
−10.37471786151840045691363249576, −9.210784312348211499329292040172, −8.226960023402415199990449464960, −6.75993562858045550226437068520, −6.28506110156148217916571279283, −5.31242711108693296454792329078, −4.52635305427063206545387281708, −4.12988184856331408304687946418, −1.72753580595219942442770593971, 0,
1.72753580595219942442770593971, 4.12988184856331408304687946418, 4.52635305427063206545387281708, 5.31242711108693296454792329078, 6.28506110156148217916571279283, 6.75993562858045550226437068520, 8.226960023402415199990449464960, 9.210784312348211499329292040172, 10.37471786151840045691363249576
