| L(s) = 1 | − 1.45·3-s − 4.53·5-s + 7·7-s − 24.8·9-s + 11·11-s + 87.6·13-s + 6.61·15-s + 88.4·17-s − 55.2·19-s − 10.2·21-s − 22.8·23-s − 104.·25-s + 75.6·27-s − 258.·29-s + 194.·31-s − 16.0·33-s − 31.7·35-s − 295.·37-s − 127.·39-s − 157.·41-s + 305.·43-s + 112.·45-s − 187.·47-s + 49·49-s − 128.·51-s − 493.·53-s − 49.9·55-s + ⋯ |
| L(s) = 1 | − 0.280·3-s − 0.406·5-s + 0.377·7-s − 0.921·9-s + 0.301·11-s + 1.86·13-s + 0.113·15-s + 1.26·17-s − 0.667·19-s − 0.106·21-s − 0.207·23-s − 0.835·25-s + 0.538·27-s − 1.65·29-s + 1.12·31-s − 0.0845·33-s − 0.153·35-s − 1.31·37-s − 0.524·39-s − 0.600·41-s + 1.08·43-s + 0.374·45-s − 0.583·47-s + 0.142·49-s − 0.353·51-s − 1.28·53-s − 0.122·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.721675034\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.721675034\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - 7T \) |
| 11 | \( 1 - 11T \) |
| good | 3 | \( 1 + 1.45T + 27T^{2} \) |
| 5 | \( 1 + 4.53T + 125T^{2} \) |
| 13 | \( 1 - 87.6T + 2.19e3T^{2} \) |
| 17 | \( 1 - 88.4T + 4.91e3T^{2} \) |
| 19 | \( 1 + 55.2T + 6.85e3T^{2} \) |
| 23 | \( 1 + 22.8T + 1.21e4T^{2} \) |
| 29 | \( 1 + 258.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 194.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 295.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 157.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 305.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 187.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 493.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 378.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 714.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 647.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 1.02e3T + 3.57e5T^{2} \) |
| 73 | \( 1 - 346.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.09e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.11e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 63.3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 987.T + 9.12e5T^{2} \) |
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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
−9.241716277153249397605002121363, −8.321449623716359427753082331776, −8.016870457932428109614736476559, −6.73657125629182542123105477838, −5.92517998335379695243531034963, −5.29111251817039556946529945746, −3.94796875053655742253127394719, −3.38319359169474144126795626761, −1.87124062176914458416527146075, −0.69036368070218480660008768501,
0.69036368070218480660008768501, 1.87124062176914458416527146075, 3.38319359169474144126795626761, 3.94796875053655742253127394719, 5.29111251817039556946529945746, 5.92517998335379695243531034963, 6.73657125629182542123105477838, 8.016870457932428109614736476559, 8.321449623716359427753082331776, 9.241716277153249397605002121363
