L(s) = 1 | + 2-s + 4-s − 2·5-s + 7-s + 8-s − 2·10-s + 4·11-s − 4·13-s + 14-s + 16-s + 4·17-s − 2·19-s − 2·20-s + 4·22-s − 23-s − 25-s − 4·26-s + 28-s + 6·29-s + 6·31-s + 32-s + 4·34-s − 2·35-s − 2·37-s − 2·38-s − 2·40-s + 10·41-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.894·5-s + 0.377·7-s + 0.353·8-s − 0.632·10-s + 1.20·11-s − 1.10·13-s + 0.267·14-s + 1/4·16-s + 0.970·17-s − 0.458·19-s − 0.447·20-s + 0.852·22-s − 0.208·23-s − 1/5·25-s − 0.784·26-s + 0.188·28-s + 1.11·29-s + 1.07·31-s + 0.176·32-s + 0.685·34-s − 0.338·35-s − 0.328·37-s − 0.324·38-s − 0.316·40-s + 1.56·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2898 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2898 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.711210765\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.711210765\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 23 | \( 1 + T \) |
good | 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 10 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 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
−8.571321321716148417122909683208, −7.88795325056025987657891193751, −7.24490800139193929221096821667, −6.48311905122979951029006016159, −5.64384097293196428736968900766, −4.62457588721377785584217918649, −4.17919756083774189926828941515, −3.28442513402038135337548479262, −2.27013855713297577192480035950, −0.940272139574865042463366049549,
0.940272139574865042463366049549, 2.27013855713297577192480035950, 3.28442513402038135337548479262, 4.17919756083774189926828941515, 4.62457588721377785584217918649, 5.64384097293196428736968900766, 6.48311905122979951029006016159, 7.24490800139193929221096821667, 7.88795325056025987657891193751, 8.571321321716148417122909683208