| L(s) = 1 | + 2·3-s − 4·5-s + 7-s + 9-s − 8·15-s − 2·17-s + 2·19-s + 2·21-s − 8·23-s + 11·25-s − 4·27-s − 2·29-s + 4·31-s − 4·35-s − 6·37-s + 2·41-s + 8·43-s − 4·45-s − 4·47-s + 49-s − 4·51-s + 10·53-s + 4·57-s − 6·59-s − 4·61-s + 63-s + 12·67-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 1.78·5-s + 0.377·7-s + 1/3·9-s − 2.06·15-s − 0.485·17-s + 0.458·19-s + 0.436·21-s − 1.66·23-s + 11/5·25-s − 0.769·27-s − 0.371·29-s + 0.718·31-s − 0.676·35-s − 0.986·37-s + 0.312·41-s + 1.21·43-s − 0.596·45-s − 0.583·47-s + 1/7·49-s − 0.560·51-s + 1.37·53-s + 0.529·57-s − 0.781·59-s − 0.512·61-s + 0.125·63-s + 1.46·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75712 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 5 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 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
−14.21007225171348, −14.00217246917797, −13.47495211187243, −12.72815169214447, −12.21753043433032, −11.90958307835926, −11.30435438326590, −10.95964090184147, −10.30532629943327, −9.615056316230468, −9.129883453051462, −8.500761630282227, −8.159190302894143, −7.879032596013268, −7.309290693681612, −6.855141911465804, −6.040013790482587, −5.323187524315624, −4.607108256523330, −4.023065745798761, −3.757106453055410, −3.126891272312981, −2.470738055511023, −1.854537899533384, −0.8215395710824648, 0,
0.8215395710824648, 1.854537899533384, 2.470738055511023, 3.126891272312981, 3.757106453055410, 4.023065745798761, 4.607108256523330, 5.323187524315624, 6.040013790482587, 6.855141911465804, 7.309290693681612, 7.879032596013268, 8.159190302894143, 8.500761630282227, 9.129883453051462, 9.615056316230468, 10.30532629943327, 10.95964090184147, 11.30435438326590, 11.90958307835926, 12.21753043433032, 12.72815169214447, 13.47495211187243, 14.00217246917797, 14.21007225171348
