| L(s) = 1 | − 105. i·2-s − 6.56e3i·3-s + 1.19e5·4-s − 6.92e5·6-s − 2.39e7i·7-s − 2.64e7i·8-s − 4.30e7·9-s − 5.09e8·11-s − 7.86e8i·12-s − 4.71e9i·13-s − 2.52e9·14-s + 1.29e10·16-s − 4.44e10i·17-s + 4.54e9i·18-s + 4.25e10·19-s + ⋯ |
| L(s) = 1 | − 0.291i·2-s − 0.577i·3-s + 0.915·4-s − 0.168·6-s − 1.56i·7-s − 0.558i·8-s − 0.333·9-s − 0.716·11-s − 0.528i·12-s − 1.60i·13-s − 0.457·14-s + 0.752·16-s − 1.54i·17-s + 0.0971i·18-s + 0.574·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(18-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+17/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(9)\) |
\(\approx\) |
\(2.508677973\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.508677973\) |
| \(L(\frac{19}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 6.56e3iT \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + 105. iT - 1.31e5T^{2} \) |
| 7 | \( 1 + 2.39e7iT - 2.32e14T^{2} \) |
| 11 | \( 1 + 5.09e8T + 5.05e17T^{2} \) |
| 13 | \( 1 + 4.71e9iT - 8.65e18T^{2} \) |
| 17 | \( 1 + 4.44e10iT - 8.27e20T^{2} \) |
| 19 | \( 1 - 4.25e10T + 5.48e21T^{2} \) |
| 23 | \( 1 + 4.70e11iT - 1.41e23T^{2} \) |
| 29 | \( 1 + 3.15e12T + 7.25e24T^{2} \) |
| 31 | \( 1 - 3.46e12T + 2.25e25T^{2} \) |
| 37 | \( 1 - 3.18e13iT - 4.56e26T^{2} \) |
| 41 | \( 1 - 8.13e13T + 2.61e27T^{2} \) |
| 43 | \( 1 - 1.24e14iT - 5.87e27T^{2} \) |
| 47 | \( 1 + 7.05e13iT - 2.66e28T^{2} \) |
| 53 | \( 1 + 2.51e14iT - 2.05e29T^{2} \) |
| 59 | \( 1 - 6.39e14T + 1.27e30T^{2} \) |
| 61 | \( 1 + 4.96e13T + 2.24e30T^{2} \) |
| 67 | \( 1 + 2.75e15iT - 1.10e31T^{2} \) |
| 71 | \( 1 - 2.50e15T + 2.96e31T^{2} \) |
| 73 | \( 1 + 8.85e14iT - 4.74e31T^{2} \) |
| 79 | \( 1 - 6.85e13T + 1.81e32T^{2} \) |
| 83 | \( 1 + 3.38e16iT - 4.21e32T^{2} \) |
| 89 | \( 1 - 3.37e16T + 1.37e33T^{2} \) |
| 97 | \( 1 - 6.48e16iT - 5.95e33T^{2} \) |
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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
−10.66192631682548445759312732996, −9.903537870620825947160989105141, −7.86519202130812514274940919207, −7.42812374339715194708005283989, −6.33004385670444114370674033182, −4.93359487275777056436062158175, −3.30767126327739125256275581772, −2.53803765802789144191743955701, −0.984488203538739914833421419143, −0.52152055728567989124971167506,
1.76601857712670105248704617387, 2.47930619717334067538254912113, 3.83702804165021191562603866511, 5.44911598262277834508412415450, 6.01009034871899616961011131889, 7.40932931850788025440962477724, 8.619460121600885484173959774027, 9.538980795780019933539854939508, 10.91619065849252336536892370398, 11.68570204858601322392259095683
