| L(s) = 1 | − i·3-s − 0.511i·7-s − 9-s − 1.82·11-s − 6.12i·13-s + 2.58i·17-s + 4.86·19-s − 0.511·21-s + 6.63i·23-s + i·27-s + 7.99·29-s − 4.88·31-s + 1.82i·33-s − 7.43i·37-s − 6.12·39-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s − 0.193i·7-s − 0.333·9-s − 0.550·11-s − 1.69i·13-s + 0.626i·17-s + 1.11·19-s − 0.111·21-s + 1.38i·23-s + 0.192i·27-s + 1.48·29-s − 0.877·31-s + 0.318i·33-s − 1.22i·37-s − 0.980·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7500 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.804490055\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.804490055\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 0.511iT - 7T^{2} \) |
| 11 | \( 1 + 1.82T + 11T^{2} \) |
| 13 | \( 1 + 6.12iT - 13T^{2} \) |
| 17 | \( 1 - 2.58iT - 17T^{2} \) |
| 19 | \( 1 - 4.86T + 19T^{2} \) |
| 23 | \( 1 - 6.63iT - 23T^{2} \) |
| 29 | \( 1 - 7.99T + 29T^{2} \) |
| 31 | \( 1 + 4.88T + 31T^{2} \) |
| 37 | \( 1 + 7.43iT - 37T^{2} \) |
| 41 | \( 1 - 5.73T + 41T^{2} \) |
| 43 | \( 1 - 2.05iT - 43T^{2} \) |
| 47 | \( 1 - 7.95iT - 47T^{2} \) |
| 53 | \( 1 + 1.32iT - 53T^{2} \) |
| 59 | \( 1 - 10.0T + 59T^{2} \) |
| 61 | \( 1 - 4.77T + 61T^{2} \) |
| 67 | \( 1 + 14.3iT - 67T^{2} \) |
| 71 | \( 1 + 5.36T + 71T^{2} \) |
| 73 | \( 1 - 3.48iT - 73T^{2} \) |
| 79 | \( 1 - 1.31T + 79T^{2} \) |
| 83 | \( 1 + 0.912iT - 83T^{2} \) |
| 89 | \( 1 + 11.7T + 89T^{2} \) |
| 97 | \( 1 + 7.39iT - 97T^{2} \) |
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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
−7.64062411752295106065480418550, −7.29599104454063768493718405940, −6.24990350252466707776967651926, −5.55087219360335493886086533276, −5.21177703864358678111423908717, −4.00515647102884555108961864380, −3.20208935147593062296760402354, −2.57304852765651932870674522858, −1.40321080822204991115960764086, −0.54268260749868070564455459361,
0.914327710670402886986720505260, 2.23385252125040264504759251815, 2.84221950045537320423766417852, 3.86270414795908852168812528193, 4.54841881815716638615839901965, 5.12986486302389697345603989154, 5.89505049379931444548057232144, 6.81713815615366668001183929202, 7.17610712308826711165760037076, 8.256077998590276196881225495381
