L(s) = 1 | + (−0.541 + 0.678i)3-s + (0.900 + 0.433i)5-s + (0.974 − 1.22i)7-s + (0.0549 + 0.240i)9-s + (−0.781 + 0.376i)15-s + (0.301 + 1.32i)21-s + (0.781 − 0.376i)23-s + (0.623 + 0.781i)25-s + (−0.974 − 0.469i)27-s + (0.623 − 0.781i)29-s + (1.40 − 0.678i)35-s + 0.445·41-s + (−1.75 + 0.846i)43-s + (−0.0549 + 0.240i)45-s + (0.347 − 1.52i)47-s + ⋯ |
L(s) = 1 | + (−0.541 + 0.678i)3-s + (0.900 + 0.433i)5-s + (0.974 − 1.22i)7-s + (0.0549 + 0.240i)9-s + (−0.781 + 0.376i)15-s + (0.301 + 1.32i)21-s + (0.781 − 0.376i)23-s + (0.623 + 0.781i)25-s + (−0.974 − 0.469i)27-s + (0.623 − 0.781i)29-s + (1.40 − 0.678i)35-s + 0.445·41-s + (−1.75 + 0.846i)43-s + (−0.0549 + 0.240i)45-s + (0.347 − 1.52i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.855 - 0.517i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.855 - 0.517i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.336317328\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.336317328\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-0.900 - 0.433i)T \) |
| 29 | \( 1 + (-0.623 + 0.781i)T \) |
good | 3 | \( 1 + (0.541 - 0.678i)T + (-0.222 - 0.974i)T^{2} \) |
| 7 | \( 1 + (-0.974 + 1.22i)T + (-0.222 - 0.974i)T^{2} \) |
| 11 | \( 1 + (0.900 + 0.433i)T^{2} \) |
| 13 | \( 1 + (0.900 + 0.433i)T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 + (0.222 - 0.974i)T^{2} \) |
| 23 | \( 1 + (-0.781 + 0.376i)T + (0.623 - 0.781i)T^{2} \) |
| 31 | \( 1 + (-0.623 - 0.781i)T^{2} \) |
| 37 | \( 1 + (0.900 - 0.433i)T^{2} \) |
| 41 | \( 1 - 0.445T + T^{2} \) |
| 43 | \( 1 + (1.75 - 0.846i)T + (0.623 - 0.781i)T^{2} \) |
| 47 | \( 1 + (-0.347 + 1.52i)T + (-0.900 - 0.433i)T^{2} \) |
| 53 | \( 1 + (-0.623 - 0.781i)T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 + (1.24 - 1.56i)T + (-0.222 - 0.974i)T^{2} \) |
| 67 | \( 1 + (-0.900 + 0.433i)T^{2} \) |
| 71 | \( 1 + (0.900 + 0.433i)T^{2} \) |
| 73 | \( 1 + (-0.623 + 0.781i)T^{2} \) |
| 79 | \( 1 + (0.900 - 0.433i)T^{2} \) |
| 83 | \( 1 + (-1.21 - 1.52i)T + (-0.222 + 0.974i)T^{2} \) |
| 89 | \( 1 + (1.12 + 0.541i)T + (0.623 + 0.781i)T^{2} \) |
| 97 | \( 1 + (0.222 - 0.974i)T^{2} \) |
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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
−9.481510769526061316349352004499, −8.476453476161558507555513525457, −7.64051740197154363766467773644, −6.91582424828178982498228438662, −6.07380339542690314132480463801, −5.10405836346289521559436615915, −4.65767799864013804810367681297, −3.72332201843994407407997748446, −2.44313815764331440093304674331, −1.29822828356425579814252719427,
1.28559529555366955272867843520, 2.00453247062659773354342850098, 3.13262712524698575902515676491, 4.70433871183854443832243995617, 5.27016864370105111746237817977, 5.97782519051321198790815570001, 6.60630397641935250875955856762, 7.55640756052351687242137440293, 8.480042411354167416424377698141, 9.038594642662521338949495813341