| L(s) = 1 | + 3·3-s − 4·5-s + 6·9-s + 11-s − 3·13-s − 12·15-s + 17-s + 2·19-s + 4·23-s + 11·25-s + 9·27-s − 8·31-s + 3·33-s + 8·37-s − 9·39-s + 10·43-s − 24·45-s + 10·47-s + 3·51-s + 3·53-s − 4·55-s + 6·57-s + 14·59-s + 8·61-s + 12·65-s − 10·67-s + 12·69-s + ⋯ |
| L(s) = 1 | + 1.73·3-s − 1.78·5-s + 2·9-s + 0.301·11-s − 0.832·13-s − 3.09·15-s + 0.242·17-s + 0.458·19-s + 0.834·23-s + 11/5·25-s + 1.73·27-s − 1.43·31-s + 0.522·33-s + 1.31·37-s − 1.44·39-s + 1.52·43-s − 3.57·45-s + 1.45·47-s + 0.420·51-s + 0.412·53-s − 0.539·55-s + 0.794·57-s + 1.82·59-s + 1.02·61-s + 1.48·65-s − 1.22·67-s + 1.44·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.595555469\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.595555469\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 3 | \( 1 - p T + p T^{2} \) |
| 5 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 - 10 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 - 14 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 + 5 T + p T^{2} \) |
| 73 | \( 1 - 16 T + p T^{2} \) |
| 79 | \( 1 - 11 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 9 T + p T^{2} \) |
| 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
−8.546135744533801374859937283605, −7.88145259888108741197353629005, −7.33219970595347634820696886920, −7.00911306004628352901119510439, −5.39416633360719924119943078408, −4.27940824929622244243786949106, −3.91279637895486486904498341505, −3.09224973436857989775104104718, −2.38679386712501414153520681691, −0.897265947653841331899552843846,
0.897265947653841331899552843846, 2.38679386712501414153520681691, 3.09224973436857989775104104718, 3.91279637895486486904498341505, 4.27940824929622244243786949106, 5.39416633360719924119943078408, 7.00911306004628352901119510439, 7.33219970595347634820696886920, 7.88145259888108741197353629005, 8.546135744533801374859937283605
