| L(s) = 1 | − 3-s − 5-s − 2·7-s + 9-s + 11-s − 13-s + 15-s + 17-s − 19-s + 2·21-s + 25-s − 27-s + 8·29-s + 9·31-s − 33-s + 2·35-s − 37-s + 39-s + 43-s − 45-s − 2·47-s − 3·49-s − 51-s − 6·53-s − 55-s + 57-s + 6·59-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 0.755·7-s + 1/3·9-s + 0.301·11-s − 0.277·13-s + 0.258·15-s + 0.242·17-s − 0.229·19-s + 0.436·21-s + 1/5·25-s − 0.192·27-s + 1.48·29-s + 1.61·31-s − 0.174·33-s + 0.338·35-s − 0.164·37-s + 0.160·39-s + 0.152·43-s − 0.149·45-s − 0.291·47-s − 3/7·49-s − 0.140·51-s − 0.824·53-s − 0.134·55-s + 0.132·57-s + 0.781·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 212160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 212160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 - 9 T + p T^{2} \) |
| 37 | \( 1 + T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 7 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - T + p T^{2} \) |
| 89 | \( 1 + T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−13.13581033335214, −12.60732443680900, −12.34524928489790, −11.84054012533406, −11.48044728085367, −10.93680408911794, −10.41273830694968, −9.927791919305769, −9.677977681222241, −9.064058264660684, −8.318539304082886, −8.173989606393636, −7.491431422687008, −6.809878473815867, −6.521945758879452, −6.245765455090565, −5.422407658672339, −4.962181776927913, −4.498208025346387, −3.869632465222398, −3.408940130331319, −2.734157427932128, −2.252120450567795, −1.235883907645534, −0.7681109666453383, 0,
0.7681109666453383, 1.235883907645534, 2.252120450567795, 2.734157427932128, 3.408940130331319, 3.869632465222398, 4.498208025346387, 4.962181776927913, 5.422407658672339, 6.245765455090565, 6.521945758879452, 6.809878473815867, 7.491431422687008, 8.173989606393636, 8.318539304082886, 9.064058264660684, 9.677977681222241, 9.927791919305769, 10.41273830694968, 10.93680408911794, 11.48044728085367, 11.84054012533406, 12.34524928489790, 12.60732443680900, 13.13581033335214
