L(s) = 1 | − 3-s − 5-s + 9-s − 13-s + 15-s − 6·17-s − 4·19-s + 25-s − 27-s + 6·29-s + 4·31-s − 2·37-s + 39-s − 2·41-s − 45-s − 7·49-s + 6·51-s + 6·53-s + 4·57-s + 4·59-s − 14·61-s + 65-s + 8·67-s − 10·73-s − 75-s + 4·79-s + 81-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.447·5-s + 1/3·9-s − 0.277·13-s + 0.258·15-s − 1.45·17-s − 0.917·19-s + 1/5·25-s − 0.192·27-s + 1.11·29-s + 0.718·31-s − 0.328·37-s + 0.160·39-s − 0.312·41-s − 0.149·45-s − 49-s + 0.840·51-s + 0.824·53-s + 0.529·57-s + 0.520·59-s − 1.79·61-s + 0.124·65-s + 0.977·67-s − 1.17·73-s − 0.115·75-s + 0.450·79-s + 1/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 188760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 188760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5995418053\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5995418053\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 11 | \( 1 \) |
| 13 | \( 1 + T \) |
good | 7 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−12.97768109416446, −12.69832165897069, −12.06612107224891, −11.74104150856392, −11.25158728168455, −10.84553017112281, −10.24573581160062, −10.07656932709080, −9.257111583090646, −8.718902149920911, −8.500723534827324, −7.782270959452463, −7.333771210425090, −6.678512444088167, −6.414110030639037, −5.983037306883935, −5.035987007487548, −4.842526413292723, −4.261482785364121, −3.822316716985345, −3.011115576552671, −2.457551067026903, −1.845153588856554, −1.067704765085995, −0.2505877294903689,
0.2505877294903689, 1.067704765085995, 1.845153588856554, 2.457551067026903, 3.011115576552671, 3.822316716985345, 4.261482785364121, 4.842526413292723, 5.035987007487548, 5.983037306883935, 6.414110030639037, 6.678512444088167, 7.333771210425090, 7.782270959452463, 8.500723534827324, 8.718902149920911, 9.257111583090646, 10.07656932709080, 10.24573581160062, 10.84553017112281, 11.25158728168455, 11.74104150856392, 12.06612107224891, 12.69832165897069, 12.97768109416446