| L(s) = 1 | − 3·3-s − 13.3·5-s − 19.3·7-s + 9·9-s + 44.8·11-s + 40.0·15-s − 43.0·17-s − 70.6·19-s + 58.0·21-s + 22.7·23-s + 52.9·25-s − 27·27-s − 33.9·29-s − 35.6·31-s − 134.·33-s + 258.·35-s + 247.·37-s + 377.·41-s − 182.·43-s − 120.·45-s + 238.·47-s + 31.8·49-s + 129.·51-s − 471.·53-s − 598.·55-s + 212.·57-s + 454.·59-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.19·5-s − 1.04·7-s + 0.333·9-s + 1.22·11-s + 0.688·15-s − 0.614·17-s − 0.853·19-s + 0.603·21-s + 0.206·23-s + 0.423·25-s − 0.192·27-s − 0.217·29-s − 0.206·31-s − 0.709·33-s + 1.24·35-s + 1.10·37-s + 1.43·41-s − 0.648·43-s − 0.397·45-s + 0.738·47-s + 0.0929·49-s + 0.354·51-s − 1.22·53-s − 1.46·55-s + 0.492·57-s + 1.00·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2028 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2028 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + 3T \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 + 13.3T + 125T^{2} \) |
| 7 | \( 1 + 19.3T + 343T^{2} \) |
| 11 | \( 1 - 44.8T + 1.33e3T^{2} \) |
| 17 | \( 1 + 43.0T + 4.91e3T^{2} \) |
| 19 | \( 1 + 70.6T + 6.85e3T^{2} \) |
| 23 | \( 1 - 22.7T + 1.21e4T^{2} \) |
| 29 | \( 1 + 33.9T + 2.43e4T^{2} \) |
| 31 | \( 1 + 35.6T + 2.97e4T^{2} \) |
| 37 | \( 1 - 247.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 377.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 182.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 238.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 471.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 454.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 840.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 174.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 892.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 861.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.26e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 498.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 572.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.44e3T + 9.12e5T^{2} \) |
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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
−8.370760918454355342644042943519, −7.54286017208855447569557018190, −6.56297426777157717358383709083, −6.40315078246067709621184923088, −5.12096115427183976576831343899, −3.97975802693994877572111821620, −3.80877487110536287744716331628, −2.42411006206536115033719856411, −0.913407107046337348621666970069, 0,
0.913407107046337348621666970069, 2.42411006206536115033719856411, 3.80877487110536287744716331628, 3.97975802693994877572111821620, 5.12096115427183976576831343899, 6.40315078246067709621184923088, 6.56297426777157717358383709083, 7.54286017208855447569557018190, 8.370760918454355342644042943519
