L(s) = 1 | + 3·3-s − 3.46·5-s + 24.2·7-s + 9·9-s + 48·11-s − 41.5·13-s − 10.3·15-s + 54·17-s − 4·19-s + 72.7·21-s + 173.·23-s − 113·25-s + 27·27-s − 162.·29-s − 58.8·31-s + 144·33-s − 84·35-s + 325.·37-s − 124.·39-s + 294·41-s + 188·43-s − 31.1·45-s − 505.·47-s + 245·49-s + 162·51-s − 744.·53-s − 166.·55-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.309·5-s + 1.30·7-s + 0.333·9-s + 1.31·11-s − 0.886·13-s − 0.178·15-s + 0.770·17-s − 0.0482·19-s + 0.755·21-s + 1.57·23-s − 0.904·25-s + 0.192·27-s − 1.04·29-s − 0.341·31-s + 0.759·33-s − 0.405·35-s + 1.44·37-s − 0.512·39-s + 1.11·41-s + 0.666·43-s − 0.103·45-s − 1.56·47-s + 0.714·49-s + 0.444·51-s − 1.93·53-s − 0.407·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(3.122806130\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.122806130\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - 3T \) |
good | 5 | \( 1 + 3.46T + 125T^{2} \) |
| 7 | \( 1 - 24.2T + 343T^{2} \) |
| 11 | \( 1 - 48T + 1.33e3T^{2} \) |
| 13 | \( 1 + 41.5T + 2.19e3T^{2} \) |
| 17 | \( 1 - 54T + 4.91e3T^{2} \) |
| 19 | \( 1 + 4T + 6.85e3T^{2} \) |
| 23 | \( 1 - 173.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 162.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 58.8T + 2.97e4T^{2} \) |
| 37 | \( 1 - 325.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 294T + 6.89e4T^{2} \) |
| 43 | \( 1 - 188T + 7.95e4T^{2} \) |
| 47 | \( 1 + 505.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 744.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 252T + 2.05e5T^{2} \) |
| 61 | \( 1 - 90.0T + 2.26e5T^{2} \) |
| 67 | \( 1 - 628T + 3.00e5T^{2} \) |
| 71 | \( 1 + 6.92T + 3.57e5T^{2} \) |
| 73 | \( 1 - 1.00e3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.34e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 720T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.48e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.82e3T + 9.12e5T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.585777157101643877839423888261, −9.232139199266545057078776861472, −7.993642063733579301703699665021, −7.65492739301406495932481225119, −6.57873635333717367301621435925, −5.27537782648544382172170985020, −4.41940794080689230665277687241, −3.46904425121685817270166890375, −2.11401839644373160514748643196, −1.05397915992800364683052286641,
1.05397915992800364683052286641, 2.11401839644373160514748643196, 3.46904425121685817270166890375, 4.41940794080689230665277687241, 5.27537782648544382172170985020, 6.57873635333717367301621435925, 7.65492739301406495932481225119, 7.993642063733579301703699665021, 9.232139199266545057078776861472, 9.585777157101643877839423888261