| L(s) = 1 | − 3.29·3-s + 3.08·5-s + 1.78·7-s + 7.87·9-s + 5.08·11-s + 1.29·13-s − 10.1·15-s + 0.213·17-s + 19-s − 5.89·21-s − 3.72·23-s + 4.51·25-s − 16.0·27-s − 0.870·29-s − 16.7·33-s + 5.51·35-s + 2·37-s − 4.27·39-s + 8.59·41-s + 3.67·43-s + 24.2·45-s − 4.65·47-s − 3.80·49-s − 0.702·51-s − 11.0·53-s + 15.6·55-s − 3.29·57-s + ⋯ |
| L(s) = 1 | − 1.90·3-s + 1.37·5-s + 0.675·7-s + 2.62·9-s + 1.53·11-s + 0.359·13-s − 2.62·15-s + 0.0517·17-s + 0.229·19-s − 1.28·21-s − 0.776·23-s + 0.902·25-s − 3.09·27-s − 0.161·29-s − 2.91·33-s + 0.931·35-s + 0.328·37-s − 0.684·39-s + 1.34·41-s + 0.560·43-s + 3.61·45-s − 0.679·47-s − 0.543·49-s − 0.0984·51-s − 1.51·53-s + 2.11·55-s − 0.436·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.416628064\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.416628064\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 + 3.29T + 3T^{2} \) |
| 5 | \( 1 - 3.08T + 5T^{2} \) |
| 7 | \( 1 - 1.78T + 7T^{2} \) |
| 11 | \( 1 - 5.08T + 11T^{2} \) |
| 13 | \( 1 - 1.29T + 13T^{2} \) |
| 17 | \( 1 - 0.213T + 17T^{2} \) |
| 23 | \( 1 + 3.72T + 23T^{2} \) |
| 29 | \( 1 + 0.870T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 2T + 37T^{2} \) |
| 41 | \( 1 - 8.59T + 41T^{2} \) |
| 43 | \( 1 - 3.67T + 43T^{2} \) |
| 47 | \( 1 + 4.65T + 47T^{2} \) |
| 53 | \( 1 + 11.0T + 53T^{2} \) |
| 59 | \( 1 - 4.70T + 59T^{2} \) |
| 61 | \( 1 + 3.51T + 61T^{2} \) |
| 67 | \( 1 + 1.12T + 67T^{2} \) |
| 71 | \( 1 - 8.76T + 71T^{2} \) |
| 73 | \( 1 - 6.80T + 73T^{2} \) |
| 79 | \( 1 - 14.5T + 79T^{2} \) |
| 83 | \( 1 - 9.74T + 83T^{2} \) |
| 89 | \( 1 + 6.76T + 89T^{2} \) |
| 97 | \( 1 - 4.16T + 97T^{2} \) |
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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
−9.627364042693296189936358866350, −9.470781073789833637093318903004, −7.963477130762389877664919544672, −6.77457981668460925208149187536, −6.25235473640743344254037613069, −5.67715545051411091771081392462, −4.85154303957576770722313795609, −3.96724631199645914019872578227, −1.86873765549236741581451181933, −1.09485038587736005961420078244,
1.09485038587736005961420078244, 1.86873765549236741581451181933, 3.96724631199645914019872578227, 4.85154303957576770722313795609, 5.67715545051411091771081392462, 6.25235473640743344254037613069, 6.77457981668460925208149187536, 7.963477130762389877664919544672, 9.470781073789833637093318903004, 9.627364042693296189936358866350
