L(s) = 1 | − 2-s + 3-s + 4-s − 2·5-s − 6-s − 7-s − 8-s + 9-s + 2·10-s − 4·11-s + 12-s + 14-s − 2·15-s + 16-s + 7·17-s − 18-s + 2·19-s − 2·20-s − 21-s + 4·22-s − 23-s − 24-s − 25-s + 27-s − 28-s − 2·29-s + 2·30-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.894·5-s − 0.408·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s + 0.632·10-s − 1.20·11-s + 0.288·12-s + 0.267·14-s − 0.516·15-s + 1/4·16-s + 1.69·17-s − 0.235·18-s + 0.458·19-s − 0.447·20-s − 0.218·21-s + 0.852·22-s − 0.208·23-s − 0.204·24-s − 1/5·25-s + 0.192·27-s − 0.188·28-s − 0.371·29-s + 0.365·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7098 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7098 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.005695630\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.005695630\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 \) |
good | 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 7 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 9 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 5 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 - 15 T + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 + 5 T + p T^{2} \) |
| 71 | \( 1 - T + p T^{2} \) |
| 73 | \( 1 - 12 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + T + p T^{2} \) |
| 89 | \( 1 - 3 T + p T^{2} \) |
| 97 | \( 1 + 16 T + p T^{2} \) |
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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
−8.077430278882877721067291568436, −7.36514691825101635365706619451, −7.02386035809942173399789134777, −5.70331301623419040527731635891, −5.33869888676873410726399430700, −4.05407257565834131352056049157, −3.42975434456605565210913934251, −2.78108747999487318820313118952, −1.76003598592714671451499583460, −0.53986517797588535079988321035,
0.53986517797588535079988321035, 1.76003598592714671451499583460, 2.78108747999487318820313118952, 3.42975434456605565210913934251, 4.05407257565834131352056049157, 5.33869888676873410726399430700, 5.70331301623419040527731635891, 7.02386035809942173399789134777, 7.36514691825101635365706619451, 8.077430278882877721067291568436