| L(s) = 1 | − 3·3-s + 8.63i·5-s + (18.2 + 3.01i)7-s + 9·9-s − 4.41i·11-s − 47.2i·13-s − 25.9i·15-s + 130. i·17-s + 44.8·19-s + (−54.8 − 9.03i)21-s + 104. i·23-s + 50.3·25-s − 27·27-s − 168.·29-s + 27.5·31-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.772i·5-s + (0.986 + 0.162i)7-s + 0.333·9-s − 0.120i·11-s − 1.00i·13-s − 0.446i·15-s + 1.86i·17-s + 0.541·19-s + (−0.569 − 0.0939i)21-s + 0.948i·23-s + 0.403·25-s − 0.192·27-s − 1.07·29-s + 0.159·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.162 - 0.986i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.162 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.886920632\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.886920632\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 3T \) |
| 7 | \( 1 + (-18.2 - 3.01i)T \) |
| good | 5 | \( 1 - 8.63iT - 125T^{2} \) |
| 11 | \( 1 + 4.41iT - 1.33e3T^{2} \) |
| 13 | \( 1 + 47.2iT - 2.19e3T^{2} \) |
| 17 | \( 1 - 130. iT - 4.91e3T^{2} \) |
| 19 | \( 1 - 44.8T + 6.85e3T^{2} \) |
| 23 | \( 1 - 104. iT - 1.21e4T^{2} \) |
| 29 | \( 1 + 168.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 27.5T + 2.97e4T^{2} \) |
| 37 | \( 1 - 425.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 332. iT - 6.89e4T^{2} \) |
| 43 | \( 1 + 218. iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 390.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 736.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 469.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 21.9iT - 2.26e5T^{2} \) |
| 67 | \( 1 - 365. iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 620. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + 938. iT - 3.89e5T^{2} \) |
| 79 | \( 1 - 883. iT - 4.93e5T^{2} \) |
| 83 | \( 1 + 739.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 66.4iT - 7.04e5T^{2} \) |
| 97 | \( 1 + 353. iT - 9.12e5T^{2} \) |
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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
−9.549644618621802158671159414465, −8.441471922313867446720386246873, −7.76179196599573597931668556209, −7.04238798649173721167174442233, −5.83774850695838040691230680189, −5.55669379230550019456468222149, −4.29836107915863062280145365195, −3.38484439533887914761233775541, −2.13481847555563580360859738223, −1.02744571714271515556896113199,
0.56798858769814687039705978271, 1.45820374885609832080432476055, 2.71024079065835235161996580202, 4.38212910429613807527313516780, 4.70030059880741001762329650928, 5.54384309227105530528628294645, 6.63043330006428952571712388187, 7.44906344527054898265765037983, 8.189015058021843082439124107970, 9.269995085552247470792627017876
