| L(s) = 1 | + i·2-s − 2.25i·3-s − 4-s + 2.25·6-s + 4.22i·7-s − i·8-s − 2.08·9-s − 5.13·11-s + 2.25i·12-s − 3.16i·13-s − 4.22·14-s + 16-s − 6.48i·17-s − 2.08i·18-s + 19-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 1.30i·3-s − 0.5·4-s + 0.920·6-s + 1.59i·7-s − 0.353i·8-s − 0.695·9-s − 1.54·11-s + 0.651i·12-s − 0.878i·13-s − 1.12·14-s + 0.250·16-s − 1.57i·17-s − 0.492i·18-s + 0.229·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.377670 - 0.611084i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.377670 - 0.611084i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - iT \) |
| 5 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 + 2.25iT - 3T^{2} \) |
| 7 | \( 1 - 4.22iT - 7T^{2} \) |
| 11 | \( 1 + 5.13T + 11T^{2} \) |
| 13 | \( 1 + 3.16iT - 13T^{2} \) |
| 17 | \( 1 + 6.48iT - 17T^{2} \) |
| 23 | \( 1 + 7.56iT - 23T^{2} \) |
| 29 | \( 1 + 0.832T + 29T^{2} \) |
| 31 | \( 1 + 4.51T + 31T^{2} \) |
| 37 | \( 1 - 0.137iT - 37T^{2} \) |
| 41 | \( 1 + 11.6T + 41T^{2} \) |
| 43 | \( 1 - 2.51iT - 43T^{2} \) |
| 47 | \( 1 + 5.96iT - 47T^{2} \) |
| 53 | \( 1 + 0.225iT - 53T^{2} \) |
| 59 | \( 1 + 5.39T + 59T^{2} \) |
| 61 | \( 1 - 14.4T + 61T^{2} \) |
| 67 | \( 1 + 4.11iT - 67T^{2} \) |
| 71 | \( 1 - 3.82T + 71T^{2} \) |
| 73 | \( 1 + 4.70iT - 73T^{2} \) |
| 79 | \( 1 + 10.6T + 79T^{2} \) |
| 83 | \( 1 - 12.0iT - 83T^{2} \) |
| 89 | \( 1 - 10T + 89T^{2} \) |
| 97 | \( 1 - 3.93iT - 97T^{2} \) |
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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
−9.548911905785228662452944478361, −8.513497742179429811919591467299, −8.075948960703245292383938897940, −7.23666778965385446736914062510, −6.45336998258886070216998797739, −5.42908427484588947009259537960, −5.07510586676770463284290604887, −2.94648943131186639605656885630, −2.24223472969681158033100665910, −0.32428825887874405672795430583,
1.67398278921335836047318653017, 3.38198976849340266348304942073, 3.91575773019297067114119461641, 4.74746201180210820991398605848, 5.60669785314984157326092472289, 7.09444369463094581363859163156, 7.922432831757393884094617447393, 8.915718838941697611021105402072, 9.995937521223160086940259846050, 10.16864395489381516778857512623
