| L(s) = 1 | − 2·3-s + 7-s + 9-s + 4·13-s − 6·17-s − 2·19-s − 2·21-s − 5·25-s + 4·27-s + 6·29-s + 4·31-s + 2·37-s − 8·39-s + 6·41-s − 8·43-s + 12·47-s + 49-s + 12·51-s + 6·53-s + 4·57-s − 6·59-s + 8·61-s + 63-s + 4·67-s + 2·73-s + 10·75-s + 8·79-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.377·7-s + 1/3·9-s + 1.10·13-s − 1.45·17-s − 0.458·19-s − 0.436·21-s − 25-s + 0.769·27-s + 1.11·29-s + 0.718·31-s + 0.328·37-s − 1.28·39-s + 0.937·41-s − 1.21·43-s + 1.75·47-s + 1/7·49-s + 1.68·51-s + 0.824·53-s + 0.529·57-s − 0.781·59-s + 1.02·61-s + 0.125·63-s + 0.488·67-s + 0.234·73-s + 1.15·75-s + 0.900·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 236992 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 236992 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.752797602\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.752797602\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−12.83716939489419, −12.30197941155099, −11.89975363726127, −11.42881012474088, −11.15895739810962, −10.60719653117612, −10.41029142419922, −9.744927334784276, −9.032449702968612, −8.738196690074722, −8.200925857554844, −7.793060316870443, −6.973392114943597, −6.607814766669401, −6.151638052500941, −5.861035017578884, −5.147155631390567, −4.779924787395413, −4.100594714313666, −3.893503986547597, −2.903419647377415, −2.353809610018927, −1.741748620469267, −0.9010821052686619, −0.4974584072505260,
0.4974584072505260, 0.9010821052686619, 1.741748620469267, 2.353809610018927, 2.903419647377415, 3.893503986547597, 4.100594714313666, 4.779924787395413, 5.147155631390567, 5.861035017578884, 6.151638052500941, 6.607814766669401, 6.973392114943597, 7.793060316870443, 8.200925857554844, 8.738196690074722, 9.032449702968612, 9.744927334784276, 10.41029142419922, 10.60719653117612, 11.15895739810962, 11.42881012474088, 11.89975363726127, 12.30197941155099, 12.83716939489419
