| L(s) = 1 | − 4·5-s − 3·9-s + 6·13-s + 2·17-s + 11·25-s − 10·29-s + 2·37-s + 8·41-s + 12·45-s − 7·49-s + 14·53-s + 12·61-s − 24·65-s + 6·73-s + 9·81-s − 8·85-s + 16·89-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 + 0.485·17-s + 11/5·25-s − 1.85·29-s + 0.328·37-s + 1.24·41-s + 1.78·45-s − 49-s + 1.92·53-s + 1.53·61-s − 2.97·65-s + 0.702·73-s + 81-s − 0.867·85-s + 1.69·89-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 92416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 92416 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.493520949\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.493520949\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 + 4 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 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 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 16 T + p T^{2} \) |
| 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
−13.90683673942610, −13.18515256671856, −12.91959798666150, −12.27435284492240, −11.66859804583418, −11.41987030224028, −11.05966526389585, −10.65349864293652, −9.889945594735198, −9.078044622878390, −8.784900420376614, −8.316479598354201, −7.735627666171734, −7.537432341809369, −6.740226011845583, −6.185233092397818, −5.598110613026770, −5.115242955531359, −4.247150359232830, −3.749646829004636, −3.533225631292412, −2.847960095859315, −2.017966219042515, −0.9793021269431089, −0.4850893015547493,
0.4850893015547493, 0.9793021269431089, 2.017966219042515, 2.847960095859315, 3.533225631292412, 3.749646829004636, 4.247150359232830, 5.115242955531359, 5.598110613026770, 6.185233092397818, 6.740226011845583, 7.537432341809369, 7.735627666171734, 8.316479598354201, 8.784900420376614, 9.078044622878390, 9.889945594735198, 10.65349864293652, 11.05966526389585, 11.41987030224028, 11.66859804583418, 12.27435284492240, 12.91959798666150, 13.18515256671856, 13.90683673942610
