L(s) = 1 | − 2-s + 4-s + 3·5-s + 7-s − 8-s − 3·10-s + 6·11-s − 13-s − 14-s + 16-s − 3·17-s + 2·19-s + 3·20-s − 6·22-s + 6·23-s + 4·25-s + 26-s + 28-s + 9·29-s − 10·31-s − 32-s + 3·34-s + 3·35-s − 7·37-s − 2·38-s − 3·40-s − 6·41-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s + 1.34·5-s + 0.377·7-s − 0.353·8-s − 0.948·10-s + 1.80·11-s − 0.277·13-s − 0.267·14-s + 1/4·16-s − 0.727·17-s + 0.458·19-s + 0.670·20-s − 1.27·22-s + 1.25·23-s + 4/5·25-s + 0.196·26-s + 0.188·28-s + 1.67·29-s − 1.79·31-s − 0.176·32-s + 0.514·34-s + 0.507·35-s − 1.15·37-s − 0.324·38-s − 0.474·40-s − 0.937·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.731889839\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.731889839\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 - T \) |
good | 5 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 11 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 7 T + p T^{2} \) |
| 79 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 3 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−9.757722057258086809536277499675, −8.934148103599289116226092539930, −8.632992281249094917428811568357, −7.05787264181528278223257612101, −6.72697856370346258141392121002, −5.71797376621720048672967665634, −4.78695796348523736572838496479, −3.40753001404160678311610708882, −2.07449184040728177750450975758, −1.25972057281136140871824724835,
1.25972057281136140871824724835, 2.07449184040728177750450975758, 3.40753001404160678311610708882, 4.78695796348523736572838496479, 5.71797376621720048672967665634, 6.72697856370346258141392121002, 7.05787264181528278223257612101, 8.632992281249094917428811568357, 8.934148103599289116226092539930, 9.757722057258086809536277499675