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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 11 | \( 1 \) |
good | 3 | \( 1 + 2 T + p T^{2} \) |
| 5 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 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
−16.28550437351760, −16.01799119730003, −15.59637933522794, −14.85938175841470, −14.36439176201800, −13.77899199881109, −12.66124793017637, −12.28689772584010, −12.05258141621111, −11.31877337686401, −10.93348194889266, −10.57899505921123, −9.644556349785255, −8.844929791524801, −8.107461508101063, −7.798355745008042, −7.113093994226523, −6.421528161280807, −5.718994079129504, −5.088185890064714, −4.348691758196620, −3.888708416279395, −3.119211061288976, −1.931287890386805, −0.7409676368246068, 0,
0.7409676368246068, 1.931287890386805, 3.119211061288976, 3.888708416279395, 4.348691758196620, 5.088185890064714, 5.718994079129504, 6.421528161280807, 7.113093994226523, 7.798355745008042, 8.107461508101063, 8.844929791524801, 9.644556349785255, 10.57899505921123, 10.93348194889266, 11.31877337686401, 12.05258141621111, 12.28689772584010, 12.66124793017637, 13.77899199881109, 14.36439176201800, 14.85938175841470, 15.59637933522794, 16.01799119730003, 16.28550437351760