L(s) = 1 | + 3-s + 7-s + 9-s + 2·11-s + 5·13-s − 3·17-s + 8·19-s + 21-s − 23-s + 27-s − 29-s + 9·31-s + 2·33-s + 2·37-s + 5·39-s + 7·41-s − 7·43-s + 12·47-s + 49-s − 3·51-s + 13·53-s + 8·57-s − 15·59-s − 7·61-s + 63-s + 4·67-s − 69-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.377·7-s + 1/3·9-s + 0.603·11-s + 1.38·13-s − 0.727·17-s + 1.83·19-s + 0.218·21-s − 0.208·23-s + 0.192·27-s − 0.185·29-s + 1.61·31-s + 0.348·33-s + 0.328·37-s + 0.800·39-s + 1.09·41-s − 1.06·43-s + 1.75·47-s + 1/7·49-s − 0.420·51-s + 1.78·53-s + 1.05·57-s − 1.95·59-s − 0.896·61-s + 0.125·63-s + 0.488·67-s − 0.120·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.419403808\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.419403808\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
good | 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 + T + p T^{2} \) |
| 31 | \( 1 - 9 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 7 T + p T^{2} \) |
| 43 | \( 1 + 7 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 13 T + p T^{2} \) |
| 59 | \( 1 + 15 T + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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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
−15.09270473114176, −14.32382631892247, −13.88817069401210, −13.53188592264000, −13.18448329264372, −12.13601736013164, −11.94280424078455, −11.30786019055014, −10.75372241082376, −10.19971756312213, −9.513727275621031, −8.946191612809251, −8.712560759473137, −7.825932839117409, −7.590769220820759, −6.761468349849791, −6.192752931075497, −5.658007396402982, −4.811489271126650, −4.227230043394901, −3.634782624178509, −3.000556910813697, −2.265588850721045, −1.337872071737065, −0.8736628957355174,
0.8736628957355174, 1.337872071737065, 2.265588850721045, 3.000556910813697, 3.634782624178509, 4.227230043394901, 4.811489271126650, 5.658007396402982, 6.192752931075497, 6.761468349849791, 7.590769220820759, 7.825932839117409, 8.712560759473137, 8.946191612809251, 9.513727275621031, 10.19971756312213, 10.75372241082376, 11.30786019055014, 11.94280424078455, 12.13601736013164, 13.18448329264372, 13.53188592264000, 13.88817069401210, 14.32382631892247, 15.09270473114176