L(s) = 1 | − 3·5-s − 7-s + 5·11-s − 7·17-s + 3·19-s − 3·23-s + 4·25-s − 7·29-s − 4·31-s + 3·35-s + 11·37-s + 4·41-s + 43-s + 8·47-s + 49-s − 2·53-s − 15·55-s + 6·59-s + 13·61-s + 4·67-s − 6·71-s + 73-s − 5·77-s + 8·79-s + 14·83-s + 21·85-s − 9·95-s + ⋯ |
L(s) = 1 | − 1.34·5-s − 0.377·7-s + 1.50·11-s − 1.69·17-s + 0.688·19-s − 0.625·23-s + 4/5·25-s − 1.29·29-s − 0.718·31-s + 0.507·35-s + 1.80·37-s + 0.624·41-s + 0.152·43-s + 1.16·47-s + 1/7·49-s − 0.274·53-s − 2.02·55-s + 0.781·59-s + 1.66·61-s + 0.488·67-s − 0.712·71-s + 0.117·73-s − 0.569·77-s + 0.900·79-s + 1.53·83-s + 2.27·85-s − 0.923·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 85176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 85176 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.278475017\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.278475017\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 \) |
good | 5 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 + 7 T + p T^{2} \) |
| 19 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + 7 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 11 T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 13 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 14 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−13.89430382604813, −13.43488886620140, −12.84955927790637, −12.42678090189525, −11.82651868843668, −11.44115699137244, −11.16206967669830, −10.66124639141415, −9.697107429514672, −9.407626799618010, −8.941002449376648, −8.408756572760755, −7.760885703133745, −7.332172728962545, −6.840095531215241, −6.310650905425697, −5.773131007965288, −4.979922444000908, −4.234503746746318, −3.875907955980250, −3.682782147519573, −2.650267714085159, −2.083453426552551, −1.116667801896625, −0.4102308447458423,
0.4102308447458423, 1.116667801896625, 2.083453426552551, 2.650267714085159, 3.682782147519573, 3.875907955980250, 4.234503746746318, 4.979922444000908, 5.773131007965288, 6.310650905425697, 6.840095531215241, 7.332172728962545, 7.760885703133745, 8.408756572760755, 8.941002449376648, 9.407626799618010, 9.697107429514672, 10.66124639141415, 11.16206967669830, 11.44115699137244, 11.82651868843668, 12.42678090189525, 12.84955927790637, 13.43488886620140, 13.89430382604813