| L(s) = 1 | − 1.65·3-s + 3.65·7-s − 0.273·9-s − 2.65·11-s − 6.13·13-s − 2.34·17-s − 19-s − 6.02·21-s + 5.48·23-s + 5.40·27-s + 0.651·29-s + 6.67·31-s + 4.37·33-s + 8.70·37-s + 10.1·39-s + 1.93·41-s − 2.65·43-s + 3.71·47-s + 6.33·49-s + 3.87·51-s − 13.7·53-s + 1.65·57-s + 7.84·59-s − 1.92·61-s − 0.999·63-s − 4.44·67-s − 9.04·69-s + ⋯ |
| L(s) = 1 | − 0.953·3-s + 1.37·7-s − 0.0912·9-s − 0.799·11-s − 1.70·13-s − 0.569·17-s − 0.229·19-s − 1.31·21-s + 1.14·23-s + 1.04·27-s + 0.120·29-s + 1.19·31-s + 0.761·33-s + 1.43·37-s + 1.62·39-s + 0.301·41-s − 0.405·43-s + 0.542·47-s + 0.904·49-s + 0.543·51-s − 1.88·53-s + 0.218·57-s + 1.02·59-s − 0.246·61-s − 0.125·63-s − 0.542·67-s − 1.08·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 + 1.65T + 3T^{2} \) |
| 7 | \( 1 - 3.65T + 7T^{2} \) |
| 11 | \( 1 + 2.65T + 11T^{2} \) |
| 13 | \( 1 + 6.13T + 13T^{2} \) |
| 17 | \( 1 + 2.34T + 17T^{2} \) |
| 23 | \( 1 - 5.48T + 23T^{2} \) |
| 29 | \( 1 - 0.651T + 29T^{2} \) |
| 31 | \( 1 - 6.67T + 31T^{2} \) |
| 37 | \( 1 - 8.70T + 37T^{2} \) |
| 41 | \( 1 - 1.93T + 41T^{2} \) |
| 43 | \( 1 + 2.65T + 43T^{2} \) |
| 47 | \( 1 - 3.71T + 47T^{2} \) |
| 53 | \( 1 + 13.7T + 53T^{2} \) |
| 59 | \( 1 - 7.84T + 59T^{2} \) |
| 61 | \( 1 + 1.92T + 61T^{2} \) |
| 67 | \( 1 + 4.44T + 67T^{2} \) |
| 71 | \( 1 + 3.54T + 71T^{2} \) |
| 73 | \( 1 - 2.48T + 73T^{2} \) |
| 79 | \( 1 - 15.1T + 79T^{2} \) |
| 83 | \( 1 + 14.7T + 83T^{2} \) |
| 89 | \( 1 + 5.06T + 89T^{2} \) |
| 97 | \( 1 + 3.22T + 97T^{2} \) |
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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
−7.59560037263763430453614314225, −6.81318119746270963512370600958, −6.07541162281930594941912676119, −5.17751350002026515101751875112, −4.88869648004747131139606027918, −4.38374489721812664341607923201, −2.84599649368881805744741691858, −2.32595351100790542565977182393, −1.10206462875254636013754300128, 0,
1.10206462875254636013754300128, 2.32595351100790542565977182393, 2.84599649368881805744741691858, 4.38374489721812664341607923201, 4.88869648004747131139606027918, 5.17751350002026515101751875112, 6.07541162281930594941912676119, 6.81318119746270963512370600958, 7.59560037263763430453614314225
