L(s) = 1 | + 4·5-s − 3·9-s + 11-s − 2·13-s + 4·17-s − 6·19-s − 4·23-s + 11·25-s − 2·29-s − 2·31-s + 10·37-s − 4·41-s + 8·43-s − 12·45-s + 2·47-s + 6·53-s + 4·55-s − 12·59-s + 14·61-s − 8·65-s + 12·67-s + 8·71-s − 4·73-s + 9·81-s − 6·83-s + 16·85-s + 6·89-s + ⋯ |
L(s) = 1 | + 1.78·5-s − 9-s + 0.301·11-s − 0.554·13-s + 0.970·17-s − 1.37·19-s − 0.834·23-s + 11/5·25-s − 0.371·29-s − 0.359·31-s + 1.64·37-s − 0.624·41-s + 1.21·43-s − 1.78·45-s + 0.291·47-s + 0.824·53-s + 0.539·55-s − 1.56·59-s + 1.79·61-s − 0.992·65-s + 1.46·67-s + 0.949·71-s − 0.468·73-s + 81-s − 0.658·83-s + 1.73·85-s + 0.635·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8624 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.677767221\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.677767221\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
good | 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 4 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−7.84243688128141633065550543669, −6.92743859216994593885061344591, −6.13084714923078385076923190925, −5.85808430856797751118865780237, −5.22279012395712767950269184635, −4.34539946912301064945419937597, −3.32636164825504490335628616164, −2.35792223982950421513090216645, −2.04253952139450276371464204879, −0.78385397243241047554565051843,
0.78385397243241047554565051843, 2.04253952139450276371464204879, 2.35792223982950421513090216645, 3.32636164825504490335628616164, 4.34539946912301064945419937597, 5.22279012395712767950269184635, 5.85808430856797751118865780237, 6.13084714923078385076923190925, 6.92743859216994593885061344591, 7.84243688128141633065550543669