L(s) = 1 | + 2-s + 3-s + 4-s + 5-s + 6-s + 7-s + 8-s + 9-s + 10-s − 1.16·11-s + 12-s − 2.26·13-s + 14-s + 15-s + 16-s + 3.43·17-s + 18-s + 5.43·19-s + 20-s + 21-s − 1.16·22-s + 23-s + 24-s + 25-s − 2.26·26-s + 27-s + 28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.447·5-s + 0.408·6-s + 0.377·7-s + 0.353·8-s + 0.333·9-s + 0.316·10-s − 0.351·11-s + 0.288·12-s − 0.628·13-s + 0.267·14-s + 0.258·15-s + 0.250·16-s + 0.832·17-s + 0.235·18-s + 1.24·19-s + 0.223·20-s + 0.218·21-s − 0.248·22-s + 0.208·23-s + 0.204·24-s + 0.200·25-s − 0.444·26-s + 0.192·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4830 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4830 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.656995641\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.656995641\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 - T \) |
good | 11 | \( 1 + 1.16T + 11T^{2} \) |
| 13 | \( 1 + 2.26T + 13T^{2} \) |
| 17 | \( 1 - 3.43T + 17T^{2} \) |
| 19 | \( 1 - 5.43T + 19T^{2} \) |
| 29 | \( 1 + 4.75T + 29T^{2} \) |
| 31 | \( 1 + 4.15T + 31T^{2} \) |
| 37 | \( 1 - 0.723T + 37T^{2} \) |
| 41 | \( 1 - 8.64T + 41T^{2} \) |
| 43 | \( 1 - 8.42T + 43T^{2} \) |
| 47 | \( 1 - 0.899T + 47T^{2} \) |
| 53 | \( 1 - 3.27T + 53T^{2} \) |
| 59 | \( 1 - 5.05T + 59T^{2} \) |
| 61 | \( 1 - 6.53T + 61T^{2} \) |
| 67 | \( 1 + 10.7T + 67T^{2} \) |
| 71 | \( 1 - 3.88T + 71T^{2} \) |
| 73 | \( 1 + 3.05T + 73T^{2} \) |
| 79 | \( 1 + 1.27T + 79T^{2} \) |
| 83 | \( 1 - 7.76T + 83T^{2} \) |
| 89 | \( 1 + 9.52T + 89T^{2} \) |
| 97 | \( 1 + 10.1T + 97T^{2} \) |
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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
−8.043809693025804786273865283175, −7.48758296163527639494358170910, −7.00858474957081491514842971449, −5.73551576983595278174577539154, −5.47593581693423171609830626520, −4.56042835257216469452613313411, −3.71667652093789809839903770668, −2.87947366649306183332789136040, −2.17640170176219671308549101056, −1.10876001711045522257394106011,
1.10876001711045522257394106011, 2.17640170176219671308549101056, 2.87947366649306183332789136040, 3.71667652093789809839903770668, 4.56042835257216469452613313411, 5.47593581693423171609830626520, 5.73551576983595278174577539154, 7.00858474957081491514842971449, 7.48758296163527639494358170910, 8.043809693025804786273865283175