| L(s) = 1 | − 3-s + 2.37·5-s + 2.52i·7-s + 9-s + 2.52i·11-s + 1.58i·13-s − 2.37·15-s − 0.372·17-s + (−4 + 1.73i)19-s − 2.52i·21-s + 1.87i·23-s + 0.627·25-s − 27-s − 3.16i·29-s + 2.74·31-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.06·5-s + 0.954i·7-s + 0.333·9-s + 0.761i·11-s + 0.439i·13-s − 0.612·15-s − 0.0902·17-s + (−0.917 + 0.397i)19-s − 0.550i·21-s + 0.391i·23-s + 0.125·25-s − 0.192·27-s − 0.588i·29-s + 0.492·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.114 - 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.114 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.01480 + 0.904368i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.01480 + 0.904368i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 19 | \( 1 + (4 - 1.73i)T \) |
| good | 5 | \( 1 - 2.37T + 5T^{2} \) |
| 7 | \( 1 - 2.52iT - 7T^{2} \) |
| 11 | \( 1 - 2.52iT - 11T^{2} \) |
| 13 | \( 1 - 1.58iT - 13T^{2} \) |
| 17 | \( 1 + 0.372T + 17T^{2} \) |
| 23 | \( 1 - 1.87iT - 23T^{2} \) |
| 29 | \( 1 + 3.16iT - 29T^{2} \) |
| 31 | \( 1 - 2.74T + 31T^{2} \) |
| 37 | \( 1 - 1.58iT - 37T^{2} \) |
| 41 | \( 1 - 6.92iT - 41T^{2} \) |
| 43 | \( 1 + 0.644iT - 43T^{2} \) |
| 47 | \( 1 - 0.939iT - 47T^{2} \) |
| 53 | \( 1 - 10.0iT - 53T^{2} \) |
| 59 | \( 1 - 4T + 59T^{2} \) |
| 61 | \( 1 + 0.372T + 61T^{2} \) |
| 67 | \( 1 + 13.4T + 67T^{2} \) |
| 71 | \( 1 - 4T + 71T^{2} \) |
| 73 | \( 1 - 13.1T + 73T^{2} \) |
| 79 | \( 1 - 6.74T + 79T^{2} \) |
| 83 | \( 1 - 3.46iT - 83T^{2} \) |
| 89 | \( 1 - 13.2iT - 89T^{2} \) |
| 97 | \( 1 + 13.2iT - 97T^{2} \) |
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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
−10.13644247009225894422598479723, −9.580577486343438308325945708128, −8.806484207979930196042878131943, −7.73193153851181645934516212547, −6.54178939955786328335063981814, −6.02733441165597819252156232452, −5.17404992056839283635588640041, −4.21989022528854330677031751615, −2.55023698786691310356092210960, −1.67793217684870770229711746270,
0.69400133832863919491635967668, 2.13583013763722260368947043397, 3.55568424868120638339187717100, 4.68607500326294385003141478683, 5.63505543411522152711353015088, 6.37993078613689624851210645357, 7.14573871654818016920316633707, 8.266516151629864801763039471102, 9.156605498159344442998401449640, 10.16459629954433888652174412440
