| L(s) = 1 | − 4·5-s − 3·9-s + 6·13-s + 11·25-s − 4·29-s − 12·37-s + 8·41-s + 12·45-s − 7·49-s + 14·53-s − 12·61-s − 24·65-s + 16·73-s + 9·81-s − 10·89-s − 8·97-s + 101-s + 103-s + 107-s + 109-s + 113-s − 18·117-s + ⋯ |
| L(s) = 1 | − 1.78·5-s − 9-s + 1.66·13-s + 11/5·25-s − 0.742·29-s − 1.97·37-s + 1.24·41-s + 1.78·45-s − 49-s + 1.92·53-s − 1.53·61-s − 2.97·65-s + 1.87·73-s + 81-s − 1.05·89-s − 0.812·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s − 1.66·117-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 18496 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 18496 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8858302824\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8858302824\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 17 | \( 1 \) |
| good | 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 + 4 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 12 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 14 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 12 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 16 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−15.73580840440486, −15.27371104655826, −14.85783209291223, −14.09276596711288, −13.68353050251541, −12.95402739557412, −12.26430005732274, −11.92241099751688, −11.17626085232433, −11.03495974850709, −10.48851829368616, −9.378651242380294, −8.789635022386751, −8.374988531902308, −7.943273551228534, −7.241670977900053, −6.640936760365595, −5.884410904087345, −5.270526557378202, −4.428144477536413, −3.650527510805436, −3.510753044712856, −2.588627398005699, −1.380756207592157, −0.4178362806971421,
0.4178362806971421, 1.380756207592157, 2.588627398005699, 3.510753044712856, 3.650527510805436, 4.428144477536413, 5.270526557378202, 5.884410904087345, 6.640936760365595, 7.241670977900053, 7.943273551228534, 8.374988531902308, 8.789635022386751, 9.378651242380294, 10.48851829368616, 11.03495974850709, 11.17626085232433, 11.92241099751688, 12.26430005732274, 12.95402739557412, 13.68353050251541, 14.09276596711288, 14.85783209291223, 15.27371104655826, 15.73580840440486
