| L(s) = 1 | + 2.39·2-s − 3-s + 3.71·4-s − 2.39·6-s + 5.11·7-s + 4.11·8-s + 9-s − 3.71·12-s − 5.43·13-s + 12.2·14-s + 2.39·16-s − 1.32·17-s + 2.39·18-s + 1.67·19-s − 5.11·21-s + 2.11·23-s − 4.11·24-s − 13.0·26-s − 27-s + 19.0·28-s + 0.782·29-s − 4.43·31-s − 2.50·32-s − 3.17·34-s + 3.71·36-s + 11.3·37-s + 3.99·38-s + ⋯ |
| L(s) = 1 | + 1.69·2-s − 0.577·3-s + 1.85·4-s − 0.976·6-s + 1.93·7-s + 1.45·8-s + 0.333·9-s − 1.07·12-s − 1.50·13-s + 3.26·14-s + 0.597·16-s − 0.321·17-s + 0.563·18-s + 0.383·19-s − 1.11·21-s + 0.439·23-s − 0.838·24-s − 2.55·26-s − 0.192·27-s + 3.59·28-s + 0.145·29-s − 0.796·31-s − 0.442·32-s − 0.544·34-s + 0.619·36-s + 1.86·37-s + 0.648·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9075 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9075 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.165554407\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.165554407\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 2.39T + 2T^{2} \) |
| 7 | \( 1 - 5.11T + 7T^{2} \) |
| 13 | \( 1 + 5.43T + 13T^{2} \) |
| 17 | \( 1 + 1.32T + 17T^{2} \) |
| 19 | \( 1 - 1.67T + 19T^{2} \) |
| 23 | \( 1 - 2.11T + 23T^{2} \) |
| 29 | \( 1 - 0.782T + 29T^{2} \) |
| 31 | \( 1 + 4.43T + 31T^{2} \) |
| 37 | \( 1 - 11.3T + 37T^{2} \) |
| 41 | \( 1 - 11.5T + 41T^{2} \) |
| 43 | \( 1 - 4.78T + 43T^{2} \) |
| 47 | \( 1 + 9.45T + 47T^{2} \) |
| 53 | \( 1 - 8.23T + 53T^{2} \) |
| 59 | \( 1 - 10.0T + 59T^{2} \) |
| 61 | \( 1 + 0.779T + 61T^{2} \) |
| 67 | \( 1 - 4.45T + 67T^{2} \) |
| 71 | \( 1 - 12.9T + 71T^{2} \) |
| 73 | \( 1 + 2.32T + 73T^{2} \) |
| 79 | \( 1 - 1.12T + 79T^{2} \) |
| 83 | \( 1 - 4T + 83T^{2} \) |
| 89 | \( 1 - 14.7T + 89T^{2} \) |
| 97 | \( 1 + 4.45T + 97T^{2} \) |
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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
−7.50363976514108022995405603501, −6.95581173212146728372018228056, −6.04536185519694121091241136421, −5.36018486644753464670944945405, −4.96315564222148764488944373585, −4.46409031578273699761569315329, −3.84543060345835344565711238010, −2.54028491785893605295895330320, −2.16707262690091524487468355930, −0.982588374629938164831453141111,
0.982588374629938164831453141111, 2.16707262690091524487468355930, 2.54028491785893605295895330320, 3.84543060345835344565711238010, 4.46409031578273699761569315329, 4.96315564222148764488944373585, 5.36018486644753464670944945405, 6.04536185519694121091241136421, 6.95581173212146728372018228056, 7.50363976514108022995405603501
