| L(s) = 1 | + (1 + 1.41i)3-s − 5-s + i·7-s + (−1.00 + 2.82i)9-s + 4.82i·11-s + 1.17i·13-s + (−1 − 1.41i)15-s − 2i·17-s + 7.65·19-s + (−1.41 + i)21-s − 3.65·23-s + 25-s + (−5.00 + 1.41i)27-s − 6.82·29-s − 1.17i·31-s + ⋯ |
| L(s) = 1 | + (0.577 + 0.816i)3-s − 0.447·5-s + 0.377i·7-s + (−0.333 + 0.942i)9-s + 1.45i·11-s + 0.324i·13-s + (−0.258 − 0.365i)15-s − 0.485i·17-s + 1.75·19-s + (−0.308 + 0.218i)21-s − 0.762·23-s + 0.200·25-s + (−0.962 + 0.272i)27-s − 1.26·29-s − 0.210i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.985 - 0.169i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.985 - 0.169i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.479733050\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.479733050\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1 - 1.41i)T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - iT \) |
| good | 11 | \( 1 - 4.82iT - 11T^{2} \) |
| 13 | \( 1 - 1.17iT - 13T^{2} \) |
| 17 | \( 1 + 2iT - 17T^{2} \) |
| 19 | \( 1 - 7.65T + 19T^{2} \) |
| 23 | \( 1 + 3.65T + 23T^{2} \) |
| 29 | \( 1 + 6.82T + 29T^{2} \) |
| 31 | \( 1 + 1.17iT - 31T^{2} \) |
| 37 | \( 1 - 8.48iT - 37T^{2} \) |
| 41 | \( 1 - 6.48iT - 41T^{2} \) |
| 43 | \( 1 - 3.17T + 43T^{2} \) |
| 47 | \( 1 - 5.17T + 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 + 4.48iT - 59T^{2} \) |
| 61 | \( 1 + 5.17iT - 61T^{2} \) |
| 67 | \( 1 + 16.1T + 67T^{2} \) |
| 71 | \( 1 - 2T + 71T^{2} \) |
| 73 | \( 1 - 0.828T + 73T^{2} \) |
| 79 | \( 1 + 17.6iT - 79T^{2} \) |
| 83 | \( 1 - 4.48iT - 83T^{2} \) |
| 89 | \( 1 - 14.4iT - 89T^{2} \) |
| 97 | \( 1 + 8.82T + 97T^{2} \) |
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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
−9.319818363762480992473299460864, −8.133338513449333548522652965332, −7.63814232559268236764584993801, −6.96506188566476668378671420062, −5.76804018984120088094530630959, −4.95284976022518730074083303111, −4.38623896939520183354889499040, −3.48995147957879962613778197822, −2.66591204170486547626062587700, −1.65277424254657208888592891137,
0.41732896375128325998718991546, 1.39061346482962634162282836771, 2.67133127106166818956694202110, 3.49619896759727115033819345966, 4.01704229995340753977542026947, 5.62211099753239395103915759716, 5.84274768829771462558140066093, 7.09210108350693109224057790113, 7.51508719379284392739511705913, 8.168967425233635425398558687360
