L(s) = 1 | + 2.23·2-s + 3-s + 3.00·4-s + 3.23·5-s + 2.23·6-s + 2.23·8-s + 9-s + 7.23·10-s + 4·11-s + 3.00·12-s − 4.47·13-s + 3.23·15-s − 0.999·16-s + 7.23·17-s + 2.23·18-s − 2.76·19-s + 9.70·20-s + 8.94·22-s + 23-s + 2.23·24-s + 5.47·25-s − 10.0·26-s + 27-s − 4.47·29-s + 7.23·30-s − 2.47·31-s − 6.70·32-s + ⋯ |
L(s) = 1 | + 1.58·2-s + 0.577·3-s + 1.50·4-s + 1.44·5-s + 0.912·6-s + 0.790·8-s + 0.333·9-s + 2.28·10-s + 1.20·11-s + 0.866·12-s − 1.24·13-s + 0.835·15-s − 0.249·16-s + 1.75·17-s + 0.527·18-s − 0.634·19-s + 2.17·20-s + 1.90·22-s + 0.208·23-s + 0.456·24-s + 1.09·25-s − 1.96·26-s + 0.192·27-s − 0.830·29-s + 1.32·30-s − 0.444·31-s − 1.18·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(7.609660530\) |
\(L(\frac12)\) |
\(\approx\) |
\(7.609660530\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 7 | \( 1 \) |
| 23 | \( 1 - T \) |
good | 2 | \( 1 - 2.23T + 2T^{2} \) |
| 5 | \( 1 - 3.23T + 5T^{2} \) |
| 11 | \( 1 - 4T + 11T^{2} \) |
| 13 | \( 1 + 4.47T + 13T^{2} \) |
| 17 | \( 1 - 7.23T + 17T^{2} \) |
| 19 | \( 1 + 2.76T + 19T^{2} \) |
| 29 | \( 1 + 4.47T + 29T^{2} \) |
| 31 | \( 1 + 2.47T + 31T^{2} \) |
| 37 | \( 1 + 4.47T + 37T^{2} \) |
| 41 | \( 1 + 6.94T + 41T^{2} \) |
| 43 | \( 1 - 7.70T + 43T^{2} \) |
| 47 | \( 1 - 4T + 47T^{2} \) |
| 53 | \( 1 + 0.763T + 53T^{2} \) |
| 59 | \( 1 + 12.9T + 59T^{2} \) |
| 61 | \( 1 - 4.47T + 61T^{2} \) |
| 67 | \( 1 - 5.23T + 67T^{2} \) |
| 71 | \( 1 + 8T + 71T^{2} \) |
| 73 | \( 1 - 10.9T + 73T^{2} \) |
| 79 | \( 1 + 3.70T + 79T^{2} \) |
| 83 | \( 1 + 4T + 83T^{2} \) |
| 89 | \( 1 + 3.23T + 89T^{2} \) |
| 97 | \( 1 - 0.472T + 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.767561685462592248338143512244, −7.52549969206030121163973792865, −6.90034496901689116257258785395, −6.09352878570690142813631688580, −5.52839020489839870721034604501, −4.85018566267584296499055752839, −3.90659104544437362828171402893, −3.16413191499077546539255538092, −2.28678528909372857884199057031, −1.54419304715736438987677788097,
1.54419304715736438987677788097, 2.28678528909372857884199057031, 3.16413191499077546539255538092, 3.90659104544437362828171402893, 4.85018566267584296499055752839, 5.52839020489839870721034604501, 6.09352878570690142813631688580, 6.90034496901689116257258785395, 7.52549969206030121163973792865, 8.767561685462592248338143512244