| L(s) = 1 | + 2.45·3-s + 0.279·5-s + 3.64·7-s + 3.01·9-s + 4.16·11-s − 0.225·13-s + 0.686·15-s + 1.76·17-s − 2.09·19-s + 8.93·21-s − 2.12·23-s − 4.92·25-s + 0.0295·27-s + 1.16·29-s + 7.55·31-s + 10.2·33-s + 1.02·35-s − 2.81·37-s − 0.553·39-s + 0.812·41-s + 7.36·43-s + 0.842·45-s + 4.04·47-s + 6.29·49-s + 4.33·51-s + 0.886·53-s + 1.16·55-s + ⋯ |
| L(s) = 1 | + 1.41·3-s + 0.125·5-s + 1.37·7-s + 1.00·9-s + 1.25·11-s − 0.0626·13-s + 0.177·15-s + 0.428·17-s − 0.481·19-s + 1.95·21-s − 0.442·23-s − 0.984·25-s + 0.00568·27-s + 0.217·29-s + 1.35·31-s + 1.77·33-s + 0.172·35-s − 0.462·37-s − 0.0886·39-s + 0.126·41-s + 1.12·43-s + 0.125·45-s + 0.589·47-s + 0.898·49-s + 0.606·51-s + 0.121·53-s + 0.157·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.273414918\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.273414918\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.45T + 3T^{2} \) |
| 5 | \( 1 - 0.279T + 5T^{2} \) |
| 7 | \( 1 - 3.64T + 7T^{2} \) |
| 11 | \( 1 - 4.16T + 11T^{2} \) |
| 13 | \( 1 + 0.225T + 13T^{2} \) |
| 17 | \( 1 - 1.76T + 17T^{2} \) |
| 19 | \( 1 + 2.09T + 19T^{2} \) |
| 23 | \( 1 + 2.12T + 23T^{2} \) |
| 29 | \( 1 - 1.16T + 29T^{2} \) |
| 31 | \( 1 - 7.55T + 31T^{2} \) |
| 37 | \( 1 + 2.81T + 37T^{2} \) |
| 41 | \( 1 - 0.812T + 41T^{2} \) |
| 43 | \( 1 - 7.36T + 43T^{2} \) |
| 47 | \( 1 - 4.04T + 47T^{2} \) |
| 53 | \( 1 - 0.886T + 53T^{2} \) |
| 59 | \( 1 - 0.749T + 59T^{2} \) |
| 61 | \( 1 + 7.77T + 61T^{2} \) |
| 67 | \( 1 - 14.9T + 67T^{2} \) |
| 71 | \( 1 + 6.01T + 71T^{2} \) |
| 73 | \( 1 + 15.3T + 73T^{2} \) |
| 79 | \( 1 - 15.4T + 79T^{2} \) |
| 83 | \( 1 + 3.28T + 83T^{2} \) |
| 89 | \( 1 + 3.17T + 89T^{2} \) |
| 97 | \( 1 + 9.57T + 97T^{2} \) |
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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
−8.317615248323547004347810341908, −7.995733667822866226811136001410, −7.23399119266331806214426944488, −6.31601745174066464135060392329, −5.41963103159672163446406339405, −4.30016835431693942554044562668, −3.97564396633901810671693790171, −2.84479756048230072600795131595, −2.00419011214507568753236684675, −1.26496938325353966462997777307,
1.26496938325353966462997777307, 2.00419011214507568753236684675, 2.84479756048230072600795131595, 3.97564396633901810671693790171, 4.30016835431693942554044562668, 5.41963103159672163446406339405, 6.31601745174066464135060392329, 7.23399119266331806214426944488, 7.995733667822866226811136001410, 8.317615248323547004347810341908
