| L(s) = 1 | − 2.40i·3-s − 2.20i·7-s − 2.78·9-s − 1.68·11-s + 6.87i·13-s + 5.86i·17-s + 0.108·19-s − 5.31·21-s − 4.40i·23-s − 0.513i·27-s + 4.25·29-s − 9.07·31-s + 4.04i·33-s − 1.74i·37-s + 16.5·39-s + ⋯ |
| L(s) = 1 | − 1.38i·3-s − 0.834i·7-s − 0.928·9-s − 0.507·11-s + 1.90i·13-s + 1.42i·17-s + 0.0248·19-s − 1.15·21-s − 0.918i·23-s − 0.0987i·27-s + 0.790·29-s − 1.62·31-s + 0.704i·33-s − 0.287i·37-s + 2.64·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.412325537\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.412325537\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + 2.40iT - 3T^{2} \) |
| 7 | \( 1 + 2.20iT - 7T^{2} \) |
| 11 | \( 1 + 1.68T + 11T^{2} \) |
| 13 | \( 1 - 6.87iT - 13T^{2} \) |
| 17 | \( 1 - 5.86iT - 17T^{2} \) |
| 19 | \( 1 - 0.108T + 19T^{2} \) |
| 23 | \( 1 + 4.40iT - 23T^{2} \) |
| 29 | \( 1 - 4.25T + 29T^{2} \) |
| 31 | \( 1 + 9.07T + 31T^{2} \) |
| 37 | \( 1 + 1.74iT - 37T^{2} \) |
| 41 | \( 1 - 7.95T + 41T^{2} \) |
| 43 | \( 1 - 8.89iT - 43T^{2} \) |
| 47 | \( 1 - 10.5iT - 47T^{2} \) |
| 53 | \( 1 - 8.51iT - 53T^{2} \) |
| 59 | \( 1 - 7.67T + 59T^{2} \) |
| 61 | \( 1 - 1.94T + 61T^{2} \) |
| 67 | \( 1 - 0.788iT - 67T^{2} \) |
| 71 | \( 1 + 5.48T + 71T^{2} \) |
| 73 | \( 1 - 11.9iT - 73T^{2} \) |
| 79 | \( 1 - 1.28T + 79T^{2} \) |
| 83 | \( 1 - 4.08iT - 83T^{2} \) |
| 89 | \( 1 - 2.37T + 89T^{2} \) |
| 97 | \( 1 + 5.70iT - 97T^{2} \) |
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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
−8.275482766501717128960074220884, −7.56838609830062879538936439046, −7.03379101006718697980338715534, −6.41099484858286476777829854740, −5.82086821730217476235371254793, −4.45663093316803427550304183717, −4.02393573374010130512224018090, −2.64809713064670485827720827647, −1.81879063099329373397517166208, −1.03686488130394859177311327608,
0.45727767090307969466642449951, 2.30033042211614620867202342961, 3.14488390758048170810231885965, 3.70959877584956445409211120042, 4.94381600606945568338396978626, 5.32210404247796694966294989365, 5.78408324917380360710216193453, 7.13823717639861220534380707361, 7.78329770045127948457860160551, 8.711667353335464571468619606713
