| L(s) = 1 | + (−1.61 + 1.17i)2-s − 2.33·3-s + (1.22 − 3.80i)4-s + 3.10·5-s + (3.77 − 2.75i)6-s + (2.50 + 7.59i)8-s − 3.54·9-s + (−5.01 + 3.65i)10-s + 4.69i·11-s + (−2.85 + 8.89i)12-s − 6.88·13-s − 7.24·15-s + (−13.0 − 9.32i)16-s − 16.9i·17-s + (5.72 − 4.17i)18-s + 26.2·19-s + ⋯ |
| L(s) = 1 | + (−0.808 + 0.589i)2-s − 0.778·3-s + (0.306 − 0.952i)4-s + 0.620·5-s + (0.629 − 0.458i)6-s + (0.313 + 0.949i)8-s − 0.393·9-s + (−0.501 + 0.365i)10-s + 0.426i·11-s + (−0.238 + 0.741i)12-s − 0.529·13-s − 0.482·15-s + (−0.812 − 0.582i)16-s − 0.999i·17-s + (0.318 − 0.232i)18-s + 1.37·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.384 - 0.923i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.384 - 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.685922 + 0.457258i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.685922 + 0.457258i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.61 - 1.17i)T \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + 2.33T + 9T^{2} \) |
| 5 | \( 1 - 3.10T + 25T^{2} \) |
| 11 | \( 1 - 4.69iT - 121T^{2} \) |
| 13 | \( 1 + 6.88T + 169T^{2} \) |
| 17 | \( 1 + 16.9iT - 289T^{2} \) |
| 19 | \( 1 - 26.2T + 361T^{2} \) |
| 23 | \( 1 - 25.8T + 529T^{2} \) |
| 29 | \( 1 - 42.2iT - 841T^{2} \) |
| 31 | \( 1 + 18.3iT - 961T^{2} \) |
| 37 | \( 1 - 49.8iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 10.7iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 24.1iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 13.6iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 6.97iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 106.T + 3.48e3T^{2} \) |
| 61 | \( 1 - 93.4T + 3.72e3T^{2} \) |
| 67 | \( 1 - 89.2iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 81.7T + 5.04e3T^{2} \) |
| 73 | \( 1 - 137. iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 13.1T + 6.24e3T^{2} \) |
| 83 | \( 1 - 2.15T + 6.88e3T^{2} \) |
| 89 | \( 1 + 101. iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 88.9iT - 9.40e3T^{2} \) |
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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
−11.29760170976989675280129321915, −10.05857601662900636979221171512, −9.588842391472596719395473024928, −8.565600457214143935725526142456, −7.33568938134684892924313965220, −6.66435600961782406925665048140, −5.45297025323438777396126063695, −5.04924753572097542938866579804, −2.69601698657499326854508767882, −1.01848985938947262022994766164,
0.66809028202279275488044209094, 2.24871488258543291524240228943, 3.58973400173147674268709274101, 5.21521278106623852452029671386, 6.14256819573795724889477102143, 7.24689127395905812539796450647, 8.320514058918444719918292054915, 9.281500138561689971996669088271, 10.08496819278965907030735762375, 10.91944335112711007680590824449
