L(s) = 1 | + 5-s − 4·7-s − 3·9-s − 6·11-s + 3·13-s + 7·17-s − 19-s − 6·23-s + 25-s + 3·29-s + 7·31-s − 4·35-s − 4·37-s + 9·41-s + 7·43-s − 3·45-s + 9·47-s + 9·49-s + 6·53-s − 6·55-s − 4·59-s − 61-s + 12·63-s + 3·65-s + 13·67-s − 16·73-s + 24·77-s + ⋯ |
L(s) = 1 | + 0.447·5-s − 1.51·7-s − 9-s − 1.80·11-s + 0.832·13-s + 1.69·17-s − 0.229·19-s − 1.25·23-s + 1/5·25-s + 0.557·29-s + 1.25·31-s − 0.676·35-s − 0.657·37-s + 1.40·41-s + 1.06·43-s − 0.447·45-s + 1.31·47-s + 9/7·49-s + 0.824·53-s − 0.809·55-s − 0.520·59-s − 0.128·61-s + 1.51·63-s + 0.372·65-s + 1.58·67-s − 1.87·73-s + 2.73·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.185885249\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.185885249\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 61 | \( 1 + T \) |
good | 3 | \( 1 + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 - 7 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 - 7 T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 - 13 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 16 T + p T^{2} \) |
| 79 | \( 1 - 5 T + p T^{2} \) |
| 83 | \( 1 - 14 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 4 T + p T^{2} \) |
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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
−8.958347078715838551997807904062, −8.146799237123242062082323003127, −7.55950222148537518424995744705, −6.35574076498474401962359565424, −5.85201655436699728294329518034, −5.33951198758177971862707858334, −3.93589594432640595490706561715, −2.98498398594298629179208753867, −2.50930577593023510954249065410, −0.66210133845320669544837504014,
0.66210133845320669544837504014, 2.50930577593023510954249065410, 2.98498398594298629179208753867, 3.93589594432640595490706561715, 5.33951198758177971862707858334, 5.85201655436699728294329518034, 6.35574076498474401962359565424, 7.55950222148537518424995744705, 8.146799237123242062082323003127, 8.958347078715838551997807904062