| L(s) = 1 | − 3-s + 5-s + 9-s − 6·13-s − 15-s − 2·17-s + 8·19-s − 8·23-s + 25-s − 27-s + 2·29-s + 4·31-s + 2·37-s + 6·39-s + 6·41-s + 4·43-s + 45-s + 8·47-s + 2·51-s − 10·53-s − 8·57-s − 4·59-s − 2·61-s − 6·65-s + 4·67-s + 8·69-s + 12·71-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 1/3·9-s − 1.66·13-s − 0.258·15-s − 0.485·17-s + 1.83·19-s − 1.66·23-s + 1/5·25-s − 0.192·27-s + 0.371·29-s + 0.718·31-s + 0.328·37-s + 0.960·39-s + 0.937·41-s + 0.609·43-s + 0.149·45-s + 1.16·47-s + 0.280·51-s − 1.37·53-s − 1.05·57-s − 0.520·59-s − 0.256·61-s − 0.744·65-s + 0.488·67-s + 0.963·69-s + 1.42·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 47040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 47040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.567073851\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.567073851\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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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
−14.32226618074867, −14.11635182902086, −13.76309008679208, −12.85837379205530, −12.54234232316143, −11.90253868039205, −11.70740582575632, −10.97879507210438, −10.32841630098197, −9.912544133862378, −9.488611851113447, −9.056194293248380, −8.059449942927441, −7.611608478438457, −7.245447556566468, −6.389768021070694, −6.052927435930096, −5.318779466034655, −4.894104851196523, −4.325224297462524, −3.553502069226833, −2.633052146258709, −2.264333438136676, −1.296003845113603, −0.4805029391069623,
0.4805029391069623, 1.296003845113603, 2.264333438136676, 2.633052146258709, 3.553502069226833, 4.325224297462524, 4.894104851196523, 5.318779466034655, 6.052927435930096, 6.389768021070694, 7.245447556566468, 7.611608478438457, 8.059449942927441, 9.056194293248380, 9.488611851113447, 9.912544133862378, 10.32841630098197, 10.97879507210438, 11.70740582575632, 11.90253868039205, 12.54234232316143, 12.85837379205530, 13.76309008679208, 14.11635182902086, 14.32226618074867
