L(s) = 1 | − 2.55·3-s − 15.7·5-s + 25.1·7-s − 20.4·9-s − 15.8·11-s + 15.4·13-s + 40.2·15-s + 56.7·17-s + 107.·19-s − 64.4·21-s + 23·23-s + 122.·25-s + 121.·27-s − 267.·29-s − 35.0·31-s + 40.5·33-s − 396.·35-s + 84.9·37-s − 39.6·39-s + 296.·41-s − 353.·43-s + 322.·45-s + 86.6·47-s + 291.·49-s − 145.·51-s − 126.·53-s + 249.·55-s + ⋯ |
L(s) = 1 | − 0.492·3-s − 1.40·5-s + 1.35·7-s − 0.757·9-s − 0.434·11-s + 0.330·13-s + 0.693·15-s + 0.810·17-s + 1.29·19-s − 0.669·21-s + 0.208·23-s + 0.982·25-s + 0.865·27-s − 1.71·29-s − 0.203·31-s + 0.213·33-s − 1.91·35-s + 0.377·37-s − 0.162·39-s + 1.13·41-s − 1.25·43-s + 1.06·45-s + 0.268·47-s + 0.849·49-s − 0.398·51-s − 0.328·53-s + 0.611·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 736 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 23 | \( 1 - 23T \) |
good | 3 | \( 1 + 2.55T + 27T^{2} \) |
| 5 | \( 1 + 15.7T + 125T^{2} \) |
| 7 | \( 1 - 25.1T + 343T^{2} \) |
| 11 | \( 1 + 15.8T + 1.33e3T^{2} \) |
| 13 | \( 1 - 15.4T + 2.19e3T^{2} \) |
| 17 | \( 1 - 56.7T + 4.91e3T^{2} \) |
| 19 | \( 1 - 107.T + 6.85e3T^{2} \) |
| 29 | \( 1 + 267.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 35.0T + 2.97e4T^{2} \) |
| 37 | \( 1 - 84.9T + 5.06e4T^{2} \) |
| 41 | \( 1 - 296.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 353.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 86.6T + 1.03e5T^{2} \) |
| 53 | \( 1 + 126.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 853.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 647.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 603.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 467.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 301.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 766.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 660.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.10e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.49e3T + 9.12e5T^{2} \) |
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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
−9.518855451499853210226008701921, −8.385947764749532679738460661665, −7.86559883935223023242807760666, −7.23068401466301959060343399269, −5.70170735139579081995071950175, −5.10301890489386640698913793664, −4.04265689658861867933350153999, −3.00677234958593858023656198310, −1.28898606549943203557104086560, 0,
1.28898606549943203557104086560, 3.00677234958593858023656198310, 4.04265689658861867933350153999, 5.10301890489386640698913793664, 5.70170735139579081995071950175, 7.23068401466301959060343399269, 7.86559883935223023242807760666, 8.385947764749532679738460661665, 9.518855451499853210226008701921