| L(s) = 1 | − 1.53·3-s − 5-s + 7-s − 0.641·9-s − 11-s − 2.37·13-s + 1.53·15-s − 3.00·17-s − 2.54·19-s − 1.53·21-s − 9.14·23-s + 25-s + 5.59·27-s − 0.542·29-s − 11.1·31-s + 1.53·33-s − 35-s − 4.07·37-s + 3.65·39-s + 12.1·41-s + 4.91·43-s + 0.641·45-s − 2.06·47-s + 49-s + 4.61·51-s + 9.55·53-s + 55-s + ⋯ |
| L(s) = 1 | − 0.886·3-s − 0.447·5-s + 0.377·7-s − 0.213·9-s − 0.301·11-s − 0.659·13-s + 0.396·15-s − 0.729·17-s − 0.583·19-s − 0.335·21-s − 1.90·23-s + 0.200·25-s + 1.07·27-s − 0.100·29-s − 1.99·31-s + 0.267·33-s − 0.169·35-s − 0.669·37-s + 0.584·39-s + 1.89·41-s + 0.749·43-s + 0.0955·45-s − 0.301·47-s + 0.142·49-s + 0.646·51-s + 1.31·53-s + 0.134·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4536281701\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4536281701\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| good | 3 | \( 1 + 1.53T + 3T^{2} \) |
| 13 | \( 1 + 2.37T + 13T^{2} \) |
| 17 | \( 1 + 3.00T + 17T^{2} \) |
| 19 | \( 1 + 2.54T + 19T^{2} \) |
| 23 | \( 1 + 9.14T + 23T^{2} \) |
| 29 | \( 1 + 0.542T + 29T^{2} \) |
| 31 | \( 1 + 11.1T + 31T^{2} \) |
| 37 | \( 1 + 4.07T + 37T^{2} \) |
| 41 | \( 1 - 12.1T + 41T^{2} \) |
| 43 | \( 1 - 4.91T + 43T^{2} \) |
| 47 | \( 1 + 2.06T + 47T^{2} \) |
| 53 | \( 1 - 9.55T + 53T^{2} \) |
| 59 | \( 1 + 3.61T + 59T^{2} \) |
| 61 | \( 1 + 2.32T + 61T^{2} \) |
| 67 | \( 1 + 13.2T + 67T^{2} \) |
| 71 | \( 1 + 13.4T + 71T^{2} \) |
| 73 | \( 1 - 12.2T + 73T^{2} \) |
| 79 | \( 1 - 1.14T + 79T^{2} \) |
| 83 | \( 1 - 17.4T + 83T^{2} \) |
| 89 | \( 1 + 12.2T + 89T^{2} \) |
| 97 | \( 1 + 0.121T + 97T^{2} \) |
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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
−7.86463051007185319986345332900, −7.45021287071321904963801685105, −6.54659142520697187819841788218, −5.84451147009638499053851049908, −5.31102782260242015833311677983, −4.43620698774617201245371394740, −3.89181479463381067776071126660, −2.65495267032620768154019447435, −1.85215060489207416500834874100, −0.35110526250601984232698987475,
0.35110526250601984232698987475, 1.85215060489207416500834874100, 2.65495267032620768154019447435, 3.89181479463381067776071126660, 4.43620698774617201245371394740, 5.31102782260242015833311677983, 5.84451147009638499053851049908, 6.54659142520697187819841788218, 7.45021287071321904963801685105, 7.86463051007185319986345332900
