| L(s) = 1 | + 1.18·3-s + 5-s − 1.59·9-s − 0.230·11-s − 2.27·13-s + 1.18·15-s + 6.53·17-s + 0.260·19-s + 8.87·23-s + 25-s − 5.44·27-s + 5.42·29-s − 4.87·31-s − 0.272·33-s − 1.15·37-s − 2.69·39-s + 4.43·41-s − 4.17·43-s − 1.59·45-s − 0.923·47-s + 7.73·51-s − 2.13·53-s − 0.230·55-s + 0.308·57-s + 5.07·59-s + 8.88·61-s − 2.27·65-s + ⋯ |
| L(s) = 1 | + 0.683·3-s + 0.447·5-s − 0.532·9-s − 0.0694·11-s − 0.630·13-s + 0.305·15-s + 1.58·17-s + 0.0596·19-s + 1.84·23-s + 0.200·25-s − 1.04·27-s + 1.00·29-s − 0.874·31-s − 0.0474·33-s − 0.189·37-s − 0.430·39-s + 0.692·41-s − 0.637·43-s − 0.238·45-s − 0.134·47-s + 1.08·51-s − 0.293·53-s − 0.0310·55-s + 0.0408·57-s + 0.660·59-s + 1.13·61-s − 0.281·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.621840132\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.621840132\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| good | 3 | \( 1 - 1.18T + 3T^{2} \) |
| 11 | \( 1 + 0.230T + 11T^{2} \) |
| 13 | \( 1 + 2.27T + 13T^{2} \) |
| 17 | \( 1 - 6.53T + 17T^{2} \) |
| 19 | \( 1 - 0.260T + 19T^{2} \) |
| 23 | \( 1 - 8.87T + 23T^{2} \) |
| 29 | \( 1 - 5.42T + 29T^{2} \) |
| 31 | \( 1 + 4.87T + 31T^{2} \) |
| 37 | \( 1 + 1.15T + 37T^{2} \) |
| 41 | \( 1 - 4.43T + 41T^{2} \) |
| 43 | \( 1 + 4.17T + 43T^{2} \) |
| 47 | \( 1 + 0.923T + 47T^{2} \) |
| 53 | \( 1 + 2.13T + 53T^{2} \) |
| 59 | \( 1 - 5.07T + 59T^{2} \) |
| 61 | \( 1 - 8.88T + 61T^{2} \) |
| 67 | \( 1 + 8.15T + 67T^{2} \) |
| 71 | \( 1 - 9.06T + 71T^{2} \) |
| 73 | \( 1 - 6.00T + 73T^{2} \) |
| 79 | \( 1 + 0.112T + 79T^{2} \) |
| 83 | \( 1 + 8.72T + 83T^{2} \) |
| 89 | \( 1 - 15.9T + 89T^{2} \) |
| 97 | \( 1 - 4.29T + 97T^{2} \) |
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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
−8.564835177248795164398812834567, −7.73450805899226849024161659731, −7.18086646165789098458977844173, −6.23498065507950702051601068532, −5.38138677550356775741188374995, −4.87849174985891845520854061899, −3.54478230772689711873937403038, −3.00047224673161124470679969896, −2.14858716271025459932986164317, −0.918599883763213567705693806145,
0.918599883763213567705693806145, 2.14858716271025459932986164317, 3.00047224673161124470679969896, 3.54478230772689711873937403038, 4.87849174985891845520854061899, 5.38138677550356775741188374995, 6.23498065507950702051601068532, 7.18086646165789098458977844173, 7.73450805899226849024161659731, 8.564835177248795164398812834567
