L(s) = 1 | + 0.314·3-s − 4.24·5-s + 1.07·7-s − 2.90·9-s + 3.34·11-s − 1.31·13-s − 1.33·15-s − 2.75·17-s + 19-s + 0.338·21-s + 9.53·23-s + 13.0·25-s − 1.85·27-s − 8.28·29-s + 3.02·31-s + 1.05·33-s − 4.57·35-s + 4.48·37-s − 0.414·39-s − 2.36·41-s − 10.5·43-s + 12.3·45-s − 3.65·47-s − 5.83·49-s − 0.866·51-s + 11.8·53-s − 14.2·55-s + ⋯ |
L(s) = 1 | + 0.181·3-s − 1.89·5-s + 0.407·7-s − 0.967·9-s + 1.00·11-s − 0.365·13-s − 0.344·15-s − 0.668·17-s + 0.229·19-s + 0.0739·21-s + 1.98·23-s + 2.60·25-s − 0.357·27-s − 1.53·29-s + 0.542·31-s + 0.183·33-s − 0.773·35-s + 0.737·37-s − 0.0663·39-s − 0.369·41-s − 1.60·43-s + 1.83·45-s − 0.533·47-s − 0.834·49-s − 0.121·51-s + 1.62·53-s − 1.91·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 19 | \( 1 - T \) |
| 79 | \( 1 + T \) |
good | 3 | \( 1 - 0.314T + 3T^{2} \) |
| 5 | \( 1 + 4.24T + 5T^{2} \) |
| 7 | \( 1 - 1.07T + 7T^{2} \) |
| 11 | \( 1 - 3.34T + 11T^{2} \) |
| 13 | \( 1 + 1.31T + 13T^{2} \) |
| 17 | \( 1 + 2.75T + 17T^{2} \) |
| 23 | \( 1 - 9.53T + 23T^{2} \) |
| 29 | \( 1 + 8.28T + 29T^{2} \) |
| 31 | \( 1 - 3.02T + 31T^{2} \) |
| 37 | \( 1 - 4.48T + 37T^{2} \) |
| 41 | \( 1 + 2.36T + 41T^{2} \) |
| 43 | \( 1 + 10.5T + 43T^{2} \) |
| 47 | \( 1 + 3.65T + 47T^{2} \) |
| 53 | \( 1 - 11.8T + 53T^{2} \) |
| 59 | \( 1 - 10.9T + 59T^{2} \) |
| 61 | \( 1 - 8.61T + 61T^{2} \) |
| 67 | \( 1 - 3.47T + 67T^{2} \) |
| 71 | \( 1 + 13.6T + 71T^{2} \) |
| 73 | \( 1 - 3.96T + 73T^{2} \) |
| 83 | \( 1 - 7.24T + 83T^{2} \) |
| 89 | \( 1 - 10.4T + 89T^{2} \) |
| 97 | \( 1 + 0.407T + 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.81225162712839840265451013724, −7.02857691766716646653058508608, −6.64850380253611370716341471174, −5.34218364352001922085090196439, −4.77971185396908089978861256440, −3.90511174851733833370294261718, −3.40863945416134198475346928804, −2.52937859402052583104151814497, −1.10929933429005821256921550193, 0,
1.10929933429005821256921550193, 2.52937859402052583104151814497, 3.40863945416134198475346928804, 3.90511174851733833370294261718, 4.77971185396908089978861256440, 5.34218364352001922085090196439, 6.64850380253611370716341471174, 7.02857691766716646653058508608, 7.81225162712839840265451013724