L(s) = 1 | + 2-s − 3-s + 4-s − 6-s + 3.93·7-s + 8-s + 9-s − 2.78·11-s − 12-s − 0.784·13-s + 3.93·14-s + 16-s − 0.938·17-s + 18-s + 7.87·19-s − 3.93·21-s − 2.78·22-s + 2·23-s − 24-s − 0.784·26-s − 27-s + 3.93·28-s − 0.938·29-s − 31-s + 32-s + 2.78·33-s − 0.938·34-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.408·6-s + 1.48·7-s + 0.353·8-s + 0.333·9-s − 0.839·11-s − 0.288·12-s − 0.217·13-s + 1.05·14-s + 0.250·16-s − 0.227·17-s + 0.235·18-s + 1.80·19-s − 0.859·21-s − 0.593·22-s + 0.417·23-s − 0.204·24-s − 0.153·26-s − 0.192·27-s + 0.744·28-s − 0.174·29-s − 0.179·31-s + 0.176·32-s + 0.484·33-s − 0.160·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.113694195\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.113694195\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 31 | \( 1 + T \) |
good | 7 | \( 1 - 3.93T + 7T^{2} \) |
| 11 | \( 1 + 2.78T + 11T^{2} \) |
| 13 | \( 1 + 0.784T + 13T^{2} \) |
| 17 | \( 1 + 0.938T + 17T^{2} \) |
| 19 | \( 1 - 7.87T + 19T^{2} \) |
| 23 | \( 1 - 2T + 23T^{2} \) |
| 29 | \( 1 + 0.938T + 29T^{2} \) |
| 37 | \( 1 + 4.78T + 37T^{2} \) |
| 41 | \( 1 - 3.50T + 41T^{2} \) |
| 43 | \( 1 - 7.72T + 43T^{2} \) |
| 47 | \( 1 - 4.56T + 47T^{2} \) |
| 53 | \( 1 + 2.66T + 53T^{2} \) |
| 59 | \( 1 + 1.56T + 59T^{2} \) |
| 61 | \( 1 + 0.784T + 61T^{2} \) |
| 67 | \( 1 - 3.56T + 67T^{2} \) |
| 71 | \( 1 - 4.44T + 71T^{2} \) |
| 73 | \( 1 - 10T + 73T^{2} \) |
| 79 | \( 1 - 2.93T + 79T^{2} \) |
| 83 | \( 1 + 0.660T + 83T^{2} \) |
| 89 | \( 1 + 11.2T + 89T^{2} \) |
| 97 | \( 1 - 1.36T + 97T^{2} \) |
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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
−7.976352514369393089834521074111, −7.56165856654899408967353721992, −6.90874485912108099380378832640, −5.78769987176593069052861620714, −5.25941173441692990796503357997, −4.84131051247101540544982923753, −3.97524518014401885211098900864, −2.91224824141887092447540917086, −1.96486297937420890853887718136, −0.951295549468762212932090809495,
0.951295549468762212932090809495, 1.96486297937420890853887718136, 2.91224824141887092447540917086, 3.97524518014401885211098900864, 4.84131051247101540544982923753, 5.25941173441692990796503357997, 5.78769987176593069052861620714, 6.90874485912108099380378832640, 7.56165856654899408967353721992, 7.976352514369393089834521074111