| L(s) = 1 | + 2·3-s − 5-s − 7-s + 9-s − 11-s − 2·13-s − 2·15-s − 17-s − 2·19-s − 2·21-s − 4·23-s + 25-s − 4·27-s + 5·29-s − 10·31-s − 2·33-s + 35-s − 2·37-s − 4·39-s + 7·41-s − 6·43-s − 45-s − 7·47-s − 6·49-s − 2·51-s − 5·53-s + 55-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.447·5-s − 0.377·7-s + 1/3·9-s − 0.301·11-s − 0.554·13-s − 0.516·15-s − 0.242·17-s − 0.458·19-s − 0.436·21-s − 0.834·23-s + 1/5·25-s − 0.769·27-s + 0.928·29-s − 1.79·31-s − 0.348·33-s + 0.169·35-s − 0.328·37-s − 0.640·39-s + 1.09·41-s − 0.914·43-s − 0.149·45-s − 1.02·47-s − 6/7·49-s − 0.280·51-s − 0.686·53-s + 0.134·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 59840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 59840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.063515846\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.063515846\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 7 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 + 7 T + p T^{2} \) |
| 53 | \( 1 + 5 T + p T^{2} \) |
| 59 | \( 1 - 9 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 13 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 3 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 3 T + p T^{2} \) |
| 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
−14.43041091820944, −13.74565061409314, −13.49684970944172, −12.70759724591239, −12.52155862771726, −11.85842763010522, −11.19602419951558, −10.78203059319431, −10.07688695510303, −9.603626769160169, −9.160257355173857, −8.569680801648797, −8.117903113774494, −7.680960715720460, −7.153416719287914, −6.487933809434637, −5.940897330518580, −5.128248110815867, −4.597149899493830, −3.847926295616750, −3.412327638067375, −2.819997708874016, −2.187918817796731, −1.609257627432886, −0.3037497878137112,
0.3037497878137112, 1.609257627432886, 2.187918817796731, 2.819997708874016, 3.412327638067375, 3.847926295616750, 4.597149899493830, 5.128248110815867, 5.940897330518580, 6.487933809434637, 7.153416719287914, 7.680960715720460, 8.117903113774494, 8.569680801648797, 9.160257355173857, 9.603626769160169, 10.07688695510303, 10.78203059319431, 11.19602419951558, 11.85842763010522, 12.52155862771726, 12.70759724591239, 13.49684970944172, 13.74565061409314, 14.43041091820944
