L(s) = 1 | − 2-s − 3-s + 4-s + 5-s + 6-s − 2·7-s − 8-s + 9-s − 10-s + 4·11-s − 12-s − 13-s + 2·14-s − 15-s + 16-s + 4·17-s − 18-s − 2·19-s + 20-s + 2·21-s − 4·22-s + 2·23-s + 24-s + 25-s + 26-s − 27-s − 2·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s − 0.755·7-s − 0.353·8-s + 1/3·9-s − 0.316·10-s + 1.20·11-s − 0.288·12-s − 0.277·13-s + 0.534·14-s − 0.258·15-s + 1/4·16-s + 0.970·17-s − 0.235·18-s − 0.458·19-s + 0.223·20-s + 0.436·21-s − 0.852·22-s + 0.417·23-s + 0.204·24-s + 1/5·25-s + 0.196·26-s − 0.192·27-s − 0.377·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8835342653\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8835342653\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 13 | \( 1 + T \) |
good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−11.21702121751910391674584411904, −10.21018500810305834142688498374, −9.620976763027980482767638198164, −8.772194400278408675663060319707, −7.50072014239365599576466271951, −6.49330656599515595141207552224, −5.92218145052084868507888987744, −4.39848825034532417015468798767, −2.87565984705891767205138792492, −1.10731363493759878565508314296,
1.10731363493759878565508314296, 2.87565984705891767205138792492, 4.39848825034532417015468798767, 5.92218145052084868507888987744, 6.49330656599515595141207552224, 7.50072014239365599576466271951, 8.772194400278408675663060319707, 9.620976763027980482767638198164, 10.21018500810305834142688498374, 11.21702121751910391674584411904