| L(s) = 1 | − 4·5-s + 11-s + 2·13-s − 2·19-s + 4·23-s + 11·25-s − 6·29-s − 10·31-s + 2·37-s + 8·41-s − 2·47-s − 6·53-s − 4·55-s − 8·59-s + 10·61-s − 8·65-s + 12·67-s + 8·71-s − 8·73-s + 8·79-s + 10·83-s − 6·89-s + 8·95-s − 10·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | − 1.78·5-s + 0.301·11-s + 0.554·13-s − 0.458·19-s + 0.834·23-s + 11/5·25-s − 1.11·29-s − 1.79·31-s + 0.328·37-s + 1.24·41-s − 0.291·47-s − 0.824·53-s − 0.539·55-s − 1.04·59-s + 1.28·61-s − 0.992·65-s + 1.46·67-s + 0.949·71-s − 0.936·73-s + 0.900·79-s + 1.09·83-s − 0.635·89-s + 0.820·95-s − 1.01·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38808 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38808 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 5 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 10 T + p T^{2} \) |
| 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
−15.17393728342936, −14.58113677552704, −14.33284886705472, −13.36560562652524, −12.84176782397053, −12.53298177850428, −11.92690865093268, −11.25445382955935, −10.99750643947394, −10.74656116415759, −9.601289529694869, −9.189125764496765, −8.658806590006317, −7.900577968718212, −7.787908560995962, −6.930030909211837, −6.674373469963284, −5.695715850013732, −5.150301408657057, −4.338343010429207, −3.888291721895468, −3.477014690440361, −2.718141432788207, −1.723261963021258, −0.8185331107517300, 0,
0.8185331107517300, 1.723261963021258, 2.718141432788207, 3.477014690440361, 3.888291721895468, 4.338343010429207, 5.150301408657057, 5.695715850013732, 6.674373469963284, 6.930030909211837, 7.787908560995962, 7.900577968718212, 8.658806590006317, 9.189125764496765, 9.601289529694869, 10.74656116415759, 10.99750643947394, 11.25445382955935, 11.92690865093268, 12.53298177850428, 12.84176782397053, 13.36560562652524, 14.33284886705472, 14.58113677552704, 15.17393728342936
