| L(s) = 1 | − 3-s − 3.46·5-s − 7-s + 9-s − 11-s + 2·13-s + 3.46·15-s + 3.46·17-s − 5.46·19-s + 21-s − 6.92·23-s + 6.99·25-s − 27-s − 6·29-s − 5.46·31-s + 33-s + 3.46·35-s − 4.92·37-s − 2·39-s + 3.46·41-s + 10.9·43-s − 3.46·45-s − 9.46·47-s + 49-s − 3.46·51-s − 0.928·53-s + 3.46·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.54·5-s − 0.377·7-s + 0.333·9-s − 0.301·11-s + 0.554·13-s + 0.894·15-s + 0.840·17-s − 1.25·19-s + 0.218·21-s − 1.44·23-s + 1.39·25-s − 0.192·27-s − 1.11·29-s − 0.981·31-s + 0.174·33-s + 0.585·35-s − 0.810·37-s − 0.320·39-s + 0.541·41-s + 1.66·43-s − 0.516·45-s − 1.38·47-s + 0.142·49-s − 0.485·51-s − 0.127·53-s + 0.467·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3696 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3696 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5076297980\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5076297980\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| good | 5 | \( 1 + 3.46T + 5T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 - 3.46T + 17T^{2} \) |
| 19 | \( 1 + 5.46T + 19T^{2} \) |
| 23 | \( 1 + 6.92T + 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 + 5.46T + 31T^{2} \) |
| 37 | \( 1 + 4.92T + 37T^{2} \) |
| 41 | \( 1 - 3.46T + 41T^{2} \) |
| 43 | \( 1 - 10.9T + 43T^{2} \) |
| 47 | \( 1 + 9.46T + 47T^{2} \) |
| 53 | \( 1 + 0.928T + 53T^{2} \) |
| 59 | \( 1 + 6.92T + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 + 14.9T + 67T^{2} \) |
| 71 | \( 1 - 12T + 71T^{2} \) |
| 73 | \( 1 + 0.535T + 73T^{2} \) |
| 79 | \( 1 - 10.9T + 79T^{2} \) |
| 83 | \( 1 + 4.39T + 83T^{2} \) |
| 89 | \( 1 + 0.928T + 89T^{2} \) |
| 97 | \( 1 - 2T + 97T^{2} \) |
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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
−8.340236387015397119632870526799, −7.74712906506769873465685595685, −7.19765589658452355337732764635, −6.23069403853037419026283787384, −5.64414248287097687816978093614, −4.55631436939942951939904426285, −3.90101998204721900331482456138, −3.31550345376119846030269661227, −1.89787463565225180122830081737, −0.41459192870486696955593075050,
0.41459192870486696955593075050, 1.89787463565225180122830081737, 3.31550345376119846030269661227, 3.90101998204721900331482456138, 4.55631436939942951939904426285, 5.64414248287097687816978093614, 6.23069403853037419026283787384, 7.19765589658452355337732764635, 7.74712906506769873465685595685, 8.340236387015397119632870526799
