L(s) = 1 | − i·3-s − 9-s + 2i·13-s + 6i·17-s − 4·19-s − 8i·23-s + i·27-s − 2·29-s + 4·31-s − 10i·37-s + 2·39-s + 2·41-s − 4i·43-s + 8i·47-s + 7·49-s + ⋯ |
L(s) = 1 | − 0.577i·3-s − 0.333·9-s + 0.554i·13-s + 1.45i·17-s − 0.917·19-s − 1.66i·23-s + 0.192i·27-s − 0.371·29-s + 0.718·31-s − 1.64i·37-s + 0.320·39-s + 0.312·41-s − 0.609i·43-s + 1.16i·47-s + 49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.709257386\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.709257386\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + iT \) |
| 5 | \( 1 \) |
good | 7 | \( 1 - 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 2iT - 13T^{2} \) |
| 17 | \( 1 - 6iT - 17T^{2} \) |
| 19 | \( 1 + 4T + 19T^{2} \) |
| 23 | \( 1 + 8iT - 23T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 + 10iT - 37T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 + 4iT - 43T^{2} \) |
| 47 | \( 1 - 8iT - 47T^{2} \) |
| 53 | \( 1 + 2iT - 53T^{2} \) |
| 59 | \( 1 - 8T + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 - 12iT - 67T^{2} \) |
| 71 | \( 1 - 8T + 71T^{2} \) |
| 73 | \( 1 - 14iT - 73T^{2} \) |
| 79 | \( 1 - 12T + 79T^{2} \) |
| 83 | \( 1 + 4iT - 83T^{2} \) |
| 89 | \( 1 - 14T + 89T^{2} \) |
| 97 | \( 1 - 2iT - 97T^{2} \) |
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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
−8.387353562682011000021912825090, −7.47545429397997204196543281533, −6.70211304414055772302196779652, −6.20564025586596702194053603207, −5.45979155950341935542166775535, −4.32592173050270852690520462394, −3.86695968236184310449696657117, −2.52938542040133861476378263603, −1.95911853262535289835240936176, −0.70125426785301186546286960587,
0.72184400977821461794514116519, 2.11212439602308011308933801779, 3.07285180778666325242484189801, 3.74430443189032450262780649491, 4.77561544837940477077115099899, 5.22987994443436947913008872473, 6.11170514390712352922690303249, 6.88530997563765861161397912607, 7.71921342123923929873102121359, 8.289424163517198021548293951838