| L(s) = 1 | − 2-s + 3-s + 4-s − 6-s − 5·7-s − 8-s + 9-s + 12-s + 2·13-s + 5·14-s + 16-s − 3·17-s − 18-s − 2·19-s − 5·21-s − 24-s − 2·26-s + 27-s − 5·28-s + 3·29-s + 2·31-s − 32-s + 3·34-s + 36-s + 7·37-s + 2·38-s + 2·39-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s − 1.88·7-s − 0.353·8-s + 1/3·9-s + 0.288·12-s + 0.554·13-s + 1.33·14-s + 1/4·16-s − 0.727·17-s − 0.235·18-s − 0.458·19-s − 1.09·21-s − 0.204·24-s − 0.392·26-s + 0.192·27-s − 0.944·28-s + 0.557·29-s + 0.359·31-s − 0.176·32-s + 0.514·34-s + 1/6·36-s + 1.15·37-s + 0.324·38-s + 0.320·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 79350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 79350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.189195509\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.189195509\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 23 | \( 1 \) |
| good | 7 | \( 1 + 5 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 7 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 15 T + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 + 3 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−13.82120943419290, −13.40785841964010, −13.08637217411041, −12.62566233254918, −12.00681906705199, −11.51812380137248, −10.82319674984061, −10.22056515010483, −10.06591339764713, −9.372918039482342, −8.985884626958567, −8.616357532272939, −7.991861162097193, −7.358621463978118, −6.788928413697208, −6.403400014361522, −6.030733518949505, −5.237081066737191, −4.251908491317076, −3.878190265259368, −3.144890700220619, −2.692459932584735, −2.151411983559815, −1.154359460873469, −0.4140865045004603,
0.4140865045004603, 1.154359460873469, 2.151411983559815, 2.692459932584735, 3.144890700220619, 3.878190265259368, 4.251908491317076, 5.237081066737191, 6.030733518949505, 6.403400014361522, 6.788928413697208, 7.358621463978118, 7.991861162097193, 8.616357532272939, 8.985884626958567, 9.372918039482342, 10.06591339764713, 10.22056515010483, 10.82319674984061, 11.51812380137248, 12.00681906705199, 12.62566233254918, 13.08637217411041, 13.40785841964010, 13.82120943419290
