L(s) = 1 | + (1.29 − 0.578i)2-s + (−1.56 − 0.751i)3-s + (1.33 − 1.49i)4-s + (−2.44 − 0.0670i)6-s + 4.28·7-s + (0.852 − 2.69i)8-s + (1.86 + 2.34i)9-s + 2.44i·11-s + (−3.19 + 1.33i)12-s + 2.71·13-s + (5.53 − 2.48i)14-s + (−0.460 − 3.97i)16-s + 1.16·17-s + (3.77 + 1.94i)18-s − 6.05·19-s + ⋯ |
L(s) = 1 | + (0.912 − 0.409i)2-s + (−0.900 − 0.433i)3-s + (0.665 − 0.746i)4-s + (−0.999 − 0.0273i)6-s + 1.61·7-s + (0.301 − 0.953i)8-s + (0.623 + 0.781i)9-s + 0.737i·11-s + (−0.923 + 0.384i)12-s + 0.752·13-s + (1.47 − 0.662i)14-s + (−0.115 − 0.993i)16-s + 0.282·17-s + (0.888 + 0.458i)18-s − 1.38·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.344 + 0.938i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.344 + 0.938i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.87510 - 1.30879i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.87510 - 1.30879i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.29 + 0.578i)T \) |
| 3 | \( 1 + (1.56 + 0.751i)T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 - 4.28T + 7T^{2} \) |
| 11 | \( 1 - 2.44iT - 11T^{2} \) |
| 13 | \( 1 - 2.71T + 13T^{2} \) |
| 17 | \( 1 - 1.16T + 17T^{2} \) |
| 19 | \( 1 + 6.05T + 19T^{2} \) |
| 23 | \( 1 + 7.55iT - 23T^{2} \) |
| 29 | \( 1 - 0.733T + 29T^{2} \) |
| 31 | \( 1 + 0.469iT - 31T^{2} \) |
| 37 | \( 1 + 1.36T + 37T^{2} \) |
| 41 | \( 1 - 4.69iT - 41T^{2} \) |
| 43 | \( 1 - 1.50iT - 43T^{2} \) |
| 47 | \( 1 - 4.07iT - 47T^{2} \) |
| 53 | \( 1 - 1.00iT - 53T^{2} \) |
| 59 | \( 1 - 1.63iT - 59T^{2} \) |
| 61 | \( 1 - 10.9iT - 61T^{2} \) |
| 67 | \( 1 + 9.97iT - 67T^{2} \) |
| 71 | \( 1 + 11.6T + 71T^{2} \) |
| 73 | \( 1 - 9.63iT - 73T^{2} \) |
| 79 | \( 1 + 3.61iT - 79T^{2} \) |
| 83 | \( 1 + 5.45T + 83T^{2} \) |
| 89 | \( 1 + 7.75iT - 89T^{2} \) |
| 97 | \( 1 - 17.1iT - 97T^{2} \) |
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
−10.78989941486644500113816143351, −10.21009540696666062729476048330, −8.589753125431769069005479070124, −7.62817922304770955822383446043, −6.62598519879820891623675619434, −5.81822605596500699759876664637, −4.71599768078233184735021675249, −4.32808160571140347571031861504, −2.30268909714447250067437783516, −1.31685093124457451312465225366,
1.67589276211129804961176080442, 3.57556353600361567001637445914, 4.44768442352260568815648241716, 5.36066068937268269112828957759, 5.96230648954543220932916904356, 7.05892868922254457045985052942, 8.085422054221720322434205539246, 8.833145980271914278690174033811, 10.42055900512696961965880151744, 11.16349408507166084748624215388