| L(s) = 1 | − i·3-s − 0.957i·7-s − 9-s − 5.41·11-s + 2.02i·13-s − 0.642i·17-s + 5.04·19-s − 0.957·21-s − 3.51i·23-s + i·27-s − 10.1·29-s + 3.69·31-s + 5.41i·33-s − 11.3i·37-s + 2.02·39-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s − 0.361i·7-s − 0.333·9-s − 1.63·11-s + 0.560i·13-s − 0.155i·17-s + 1.15·19-s − 0.208·21-s − 0.733i·23-s + 0.192i·27-s − 1.88·29-s + 0.663·31-s + 0.942i·33-s − 1.86i·37-s + 0.323·39-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
| \(L(1)\) |
\(\approx\) |
\(0.5044664482\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5044664482\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 0.957iT - 7T^{2} \) |
| 11 | \( 1 + 5.41T + 11T^{2} \) |
| 13 | \( 1 - 2.02iT - 13T^{2} \) |
| 17 | \( 1 + 0.642iT - 17T^{2} \) |
| 19 | \( 1 - 5.04T + 19T^{2} \) |
| 23 | \( 1 + 3.51iT - 23T^{2} \) |
| 29 | \( 1 + 10.1T + 29T^{2} \) |
| 31 | \( 1 - 3.69T + 31T^{2} \) |
| 37 | \( 1 + 11.3iT - 37T^{2} \) |
| 41 | \( 1 - 3.52T + 41T^{2} \) |
| 43 | \( 1 + 0.766iT - 43T^{2} \) |
| 47 | \( 1 - 4.93iT - 47T^{2} \) |
| 53 | \( 1 + 5.94iT - 53T^{2} \) |
| 59 | \( 1 + 4.71T + 59T^{2} \) |
| 61 | \( 1 - 4.34T + 61T^{2} \) |
| 67 | \( 1 - 9.51iT - 67T^{2} \) |
| 71 | \( 1 + 11.9T + 71T^{2} \) |
| 73 | \( 1 - 5.43iT - 73T^{2} \) |
| 79 | \( 1 + 11.8T + 79T^{2} \) |
| 83 | \( 1 - 1.39iT - 83T^{2} \) |
| 89 | \( 1 - 1.70T + 89T^{2} \) |
| 97 | \( 1 - 14.5iT - 97T^{2} \) |
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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
−7.78622378576193827395305377661, −7.46218210135057698647192127834, −6.89485154924467165767083321185, −5.80134718383238464782108946781, −5.46973997657701969320460663677, −4.54907012675028459528541191569, −3.70525562678729768915214983749, −2.72700266717521333605900523856, −2.13334414101813922379678610114, −0.939686933096785742444621295637,
0.13528919373590610938154560511, 1.57967769010585034830024520963, 2.77781422968570918505446839052, 3.14384477833448300217284337969, 4.16104308637104594305840034213, 5.08713144525450225394470969502, 5.47389657341579099583070751333, 6.06068116156615487205742812303, 7.25174758892583665677756508086, 7.74417830868767424757549348127
