| L(s) = 1 | + 0.254·3-s + 3.76·5-s + 2.29·7-s − 2.93·9-s + 2.49·11-s + 6.47·13-s + 0.959·15-s + 5.17·17-s − 1.78·19-s + 0.585·21-s + 1.62·23-s + 9.17·25-s − 1.51·27-s + 3.13·29-s + 2.76·31-s + 0.636·33-s + 8.64·35-s − 0.433·37-s + 1.65·39-s − 3.69·41-s − 0.958·43-s − 11.0·45-s − 1.36·47-s − 1.72·49-s + 1.31·51-s + 9.03·53-s + 9.40·55-s + ⋯ |
| L(s) = 1 | + 0.147·3-s + 1.68·5-s + 0.867·7-s − 0.978·9-s + 0.752·11-s + 1.79·13-s + 0.247·15-s + 1.25·17-s − 0.409·19-s + 0.127·21-s + 0.338·23-s + 1.83·25-s − 0.291·27-s + 0.581·29-s + 0.497·31-s + 0.110·33-s + 1.46·35-s − 0.0712·37-s + 0.264·39-s − 0.577·41-s − 0.146·43-s − 1.64·45-s − 0.198·47-s − 0.246·49-s + 0.184·51-s + 1.24·53-s + 1.26·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.142858963\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.142858963\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 503 | \( 1 - T \) |
| good | 3 | \( 1 - 0.254T + 3T^{2} \) |
| 5 | \( 1 - 3.76T + 5T^{2} \) |
| 7 | \( 1 - 2.29T + 7T^{2} \) |
| 11 | \( 1 - 2.49T + 11T^{2} \) |
| 13 | \( 1 - 6.47T + 13T^{2} \) |
| 17 | \( 1 - 5.17T + 17T^{2} \) |
| 19 | \( 1 + 1.78T + 19T^{2} \) |
| 23 | \( 1 - 1.62T + 23T^{2} \) |
| 29 | \( 1 - 3.13T + 29T^{2} \) |
| 31 | \( 1 - 2.76T + 31T^{2} \) |
| 37 | \( 1 + 0.433T + 37T^{2} \) |
| 41 | \( 1 + 3.69T + 41T^{2} \) |
| 43 | \( 1 + 0.958T + 43T^{2} \) |
| 47 | \( 1 + 1.36T + 47T^{2} \) |
| 53 | \( 1 - 9.03T + 53T^{2} \) |
| 59 | \( 1 - 7.59T + 59T^{2} \) |
| 61 | \( 1 + 7.79T + 61T^{2} \) |
| 67 | \( 1 + 4.83T + 67T^{2} \) |
| 71 | \( 1 + 2.52T + 71T^{2} \) |
| 73 | \( 1 + 4.72T + 73T^{2} \) |
| 79 | \( 1 + 14.4T + 79T^{2} \) |
| 83 | \( 1 + 15.5T + 83T^{2} \) |
| 89 | \( 1 - 2.46T + 89T^{2} \) |
| 97 | \( 1 - 3.53T + 97T^{2} \) |
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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
−8.101407239578992703991627303656, −6.96013356253199968158515160399, −6.25578426841796773667667268449, −5.76956625443420442433069568685, −5.31033518533404898646374487196, −4.32641585410707050370679732626, −3.36205088818544777624571800175, −2.63537995715452840975846808902, −1.57033509907957605324342195208, −1.18420938061628596970456168515,
1.18420938061628596970456168515, 1.57033509907957605324342195208, 2.63537995715452840975846808902, 3.36205088818544777624571800175, 4.32641585410707050370679732626, 5.31033518533404898646374487196, 5.76956625443420442433069568685, 6.25578426841796773667667268449, 6.96013356253199968158515160399, 8.101407239578992703991627303656
