| L(s) = 1 | − 3-s + 3·5-s − 2·7-s − 2·9-s − 11-s + 2·13-s − 3·15-s − 6·17-s + 2·21-s + 23-s + 4·25-s + 5·27-s + 5·29-s + 2·31-s + 33-s − 6·35-s − 10·37-s − 2·39-s + 5·43-s − 6·45-s − 11·47-s − 3·49-s + 6·51-s + 14·53-s − 3·55-s − 5·59-s − 2·61-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.34·5-s − 0.755·7-s − 2/3·9-s − 0.301·11-s + 0.554·13-s − 0.774·15-s − 1.45·17-s + 0.436·21-s + 0.208·23-s + 4/5·25-s + 0.962·27-s + 0.928·29-s + 0.359·31-s + 0.174·33-s − 1.01·35-s − 1.64·37-s − 0.320·39-s + 0.762·43-s − 0.894·45-s − 1.60·47-s − 3/7·49-s + 0.840·51-s + 1.92·53-s − 0.404·55-s − 0.650·59-s − 0.256·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 66424 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 66424 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.110391588\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.110391588\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 - 3 T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 5 T + p T^{2} \) |
| 47 | \( 1 + 11 T + p T^{2} \) |
| 53 | \( 1 - 14 T + p T^{2} \) |
| 59 | \( 1 + 5 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 17 T + p T^{2} \) |
| 79 | \( 1 - 9 T + p T^{2} \) |
| 83 | \( 1 + 5 T + p T^{2} \) |
| 89 | \( 1 - T + p T^{2} \) |
| 97 | \( 1 - T + p T^{2} \) |
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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
−14.09811877132393, −13.42536348325936, −13.41243450447655, −12.85539304503345, −12.06146997336146, −11.82986520522101, −10.93410395624131, −10.73971925267762, −10.16734546786264, −9.720522817253858, −8.999549900018367, −8.759156522209834, −8.211925327743338, −7.218555934200973, −6.691815081155350, −6.275079757624487, −5.922858966637128, −5.312910641447896, −4.812228876379337, −4.115563198664999, −3.159935114354219, −2.774285429677117, −2.056837131738711, −1.363909154141592, −0.3599550281598627,
0.3599550281598627, 1.363909154141592, 2.056837131738711, 2.774285429677117, 3.159935114354219, 4.115563198664999, 4.812228876379337, 5.312910641447896, 5.922858966637128, 6.275079757624487, 6.691815081155350, 7.218555934200973, 8.211925327743338, 8.759156522209834, 8.999549900018367, 9.720522817253858, 10.16734546786264, 10.73971925267762, 10.93410395624131, 11.82986520522101, 12.06146997336146, 12.85539304503345, 13.41243450447655, 13.42536348325936, 14.09811877132393
