| L(s) = 1 | + 14.1i·5-s − 7i·7-s − 13.1·11-s + 30.2·13-s − 78.2i·17-s + 72.0i·19-s − 217.·23-s − 73.9·25-s − 3.49i·29-s + 305. i·31-s + 98.7·35-s + 191.·37-s + 246. i·41-s − 200. i·43-s − 338.·47-s + ⋯ |
| L(s) = 1 | + 1.26i·5-s − 0.377i·7-s − 0.359·11-s + 0.645·13-s − 1.11i·17-s + 0.870i·19-s − 1.97·23-s − 0.591·25-s − 0.0223i·29-s + 1.76i·31-s + 0.476·35-s + 0.852·37-s + 0.937i·41-s − 0.712i·43-s − 1.05·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.816 + 0.577i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.816 + 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.02212428265\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.02212428265\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + 7iT \) |
| good | 5 | \( 1 - 14.1iT - 125T^{2} \) |
| 11 | \( 1 + 13.1T + 1.33e3T^{2} \) |
| 13 | \( 1 - 30.2T + 2.19e3T^{2} \) |
| 17 | \( 1 + 78.2iT - 4.91e3T^{2} \) |
| 19 | \( 1 - 72.0iT - 6.85e3T^{2} \) |
| 23 | \( 1 + 217.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 3.49iT - 2.43e4T^{2} \) |
| 31 | \( 1 - 305. iT - 2.97e4T^{2} \) |
| 37 | \( 1 - 191.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 246. iT - 6.89e4T^{2} \) |
| 43 | \( 1 + 200. iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 338.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 423. iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 366.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 730.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 1.02e3iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 79.3T + 3.57e5T^{2} \) |
| 73 | \( 1 - 127.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 413. iT - 4.93e5T^{2} \) |
| 83 | \( 1 + 228.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 193. iT - 7.04e5T^{2} \) |
| 97 | \( 1 - 1.00e3T + 9.12e5T^{2} \) |
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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
−10.24076934577817754223558097006, −9.435415103569941831554319547047, −8.201223231488220278210762468472, −7.59264753426228194158286153192, −6.64776865152802266483859060767, −6.04709358960001247862590939206, −4.83941810209773874918344452123, −3.65914670518197624913251106483, −2.92694554368570140855843145664, −1.68077800923494327916087543220,
0.00544434334796052541368930728, 1.29140421446556347290019705666, 2.40134159948470462996059470858, 3.90749115870697522968374841100, 4.60648891282669631903653852774, 5.72636375631536368707958110526, 6.22085658378082482661731705960, 7.72996787400856062635675201164, 8.273696053723668190134124545397, 9.036059969976085093460647630983
