| L(s) = 1 | − 5-s + 4·7-s − 6·11-s + 13-s + 6·17-s − 2·19-s + 6·23-s + 25-s + 6·29-s − 2·31-s − 4·35-s + 2·37-s + 6·41-s − 2·43-s − 12·47-s + 9·49-s − 6·53-s + 6·55-s + 6·59-s + 2·61-s − 65-s + 4·67-s − 6·71-s − 10·73-s − 24·77-s + 4·79-s − 6·85-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 1.51·7-s − 1.80·11-s + 0.277·13-s + 1.45·17-s − 0.458·19-s + 1.25·23-s + 1/5·25-s + 1.11·29-s − 0.359·31-s − 0.676·35-s + 0.328·37-s + 0.937·41-s − 0.304·43-s − 1.75·47-s + 9/7·49-s − 0.824·53-s + 0.809·55-s + 0.781·59-s + 0.256·61-s − 0.124·65-s + 0.488·67-s − 0.712·71-s − 1.17·73-s − 2.73·77-s + 0.450·79-s − 0.650·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.184825631\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.184825631\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−7.71686620980824463542408497094, −7.36532752821953793491550409972, −6.30114975732985881543593056776, −5.38249054984077248840214569416, −5.01092356862613681658622064436, −4.42309936330716087844333548047, −3.32075299675776612994089632032, −2.68696908914722535236861712897, −1.67553399558326699651445373685, −0.73313466386100382839439348572,
0.73313466386100382839439348572, 1.67553399558326699651445373685, 2.68696908914722535236861712897, 3.32075299675776612994089632032, 4.42309936330716087844333548047, 5.01092356862613681658622064436, 5.38249054984077248840214569416, 6.30114975732985881543593056776, 7.36532752821953793491550409972, 7.71686620980824463542408497094
