L(s) = 1 | + 1.85·2-s − 3-s + 1.44·4-s + 3.73·5-s − 1.85·6-s − 1.02·8-s + 9-s + 6.93·10-s + 5.64·11-s − 1.44·12-s − 1.32·13-s − 3.73·15-s − 4.79·16-s + 4.03·17-s + 1.85·18-s + 7.55·19-s + 5.41·20-s + 10.4·22-s − 23-s + 1.02·24-s + 8.93·25-s − 2.46·26-s − 27-s − 9.72·29-s − 6.93·30-s − 7.95·31-s − 6.86·32-s + ⋯ |
L(s) = 1 | + 1.31·2-s − 0.577·3-s + 0.724·4-s + 1.66·5-s − 0.758·6-s − 0.361·8-s + 0.333·9-s + 2.19·10-s + 1.70·11-s − 0.418·12-s − 0.367·13-s − 0.963·15-s − 1.19·16-s + 0.979·17-s + 0.437·18-s + 1.73·19-s + 1.20·20-s + 2.23·22-s − 0.208·23-s + 0.208·24-s + 1.78·25-s − 0.482·26-s − 0.192·27-s − 1.80·29-s − 1.26·30-s − 1.42·31-s − 1.21·32-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
\(L(1)\) |
\(\approx\) |
\(4.599648586\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.599648586\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - 1.85T + 2T^{2} \) |
| 5 | \( 1 - 3.73T + 5T^{2} \) |
| 11 | \( 1 - 5.64T + 11T^{2} \) |
| 13 | \( 1 + 1.32T + 13T^{2} \) |
| 17 | \( 1 - 4.03T + 17T^{2} \) |
| 19 | \( 1 - 7.55T + 19T^{2} \) |
| 29 | \( 1 + 9.72T + 29T^{2} \) |
| 31 | \( 1 + 7.95T + 31T^{2} \) |
| 37 | \( 1 - 0.802T + 37T^{2} \) |
| 41 | \( 1 - 6.20T + 41T^{2} \) |
| 43 | \( 1 + 0.213T + 43T^{2} \) |
| 47 | \( 1 - 0.832T + 47T^{2} \) |
| 53 | \( 1 + 5.97T + 53T^{2} \) |
| 59 | \( 1 - 11.6T + 59T^{2} \) |
| 61 | \( 1 - 10.1T + 61T^{2} \) |
| 67 | \( 1 + 10.9T + 67T^{2} \) |
| 71 | \( 1 - 9.99T + 71T^{2} \) |
| 73 | \( 1 + 3.36T + 73T^{2} \) |
| 79 | \( 1 + 13.4T + 79T^{2} \) |
| 83 | \( 1 - 8.43T + 83T^{2} \) |
| 89 | \( 1 + 1.45T + 89T^{2} \) |
| 97 | \( 1 - 7.93T + 97T^{2} \) |
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
−8.960211602332483454925613905320, −7.41732760325355336006938826222, −6.81426714758284448972274227793, −5.89269604211286670558788151280, −5.66312202383044209658794987554, −5.05741287205248459238336694342, −3.97267259333569621843299216345, −3.30067444654186427024768113055, −2.10146625295548882971377706597, −1.20563037594870880208542780995,
1.20563037594870880208542780995, 2.10146625295548882971377706597, 3.30067444654186427024768113055, 3.97267259333569621843299216345, 5.05741287205248459238336694342, 5.66312202383044209658794987554, 5.89269604211286670558788151280, 6.81426714758284448972274227793, 7.41732760325355336006938826222, 8.960211602332483454925613905320