| L(s) = 1 | + 2.23·2-s + 3.00·4-s + 2.23·8-s − 0.999·16-s − 4.47·17-s + 4·19-s − 8.94·23-s + 8·31-s − 6.70·32-s − 10.0·34-s + 8.94·38-s − 20.0·46-s + 8.94·47-s − 7·49-s + 4.47·53-s + 2·61-s + 17.8·62-s − 13.0·64-s − 13.4·68-s + 12.0·76-s + 16·79-s + 17.8·83-s − 26.8·92-s + 20.0·94-s − 15.6·98-s + 10.0·106-s − 17.8·107-s + ⋯ |
| L(s) = 1 | + 1.58·2-s + 1.50·4-s + 0.790·8-s − 0.249·16-s − 1.08·17-s + 0.917·19-s − 1.86·23-s + 1.43·31-s − 1.18·32-s − 1.71·34-s + 1.45·38-s − 2.94·46-s + 1.30·47-s − 49-s + 0.614·53-s + 0.256·61-s + 2.27·62-s − 1.62·64-s − 1.62·68-s + 1.37·76-s + 1.80·79-s + 1.96·83-s − 2.79·92-s + 2.06·94-s − 1.58·98-s + 0.971·106-s − 1.72·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.582995961\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.582995961\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 - 2.23T + 2T^{2} \) |
| 7 | \( 1 + 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 + 4.47T + 17T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 + 8.94T + 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 - 8T + 31T^{2} \) |
| 37 | \( 1 + 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 - 8.94T + 47T^{2} \) |
| 53 | \( 1 - 4.47T + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 73T^{2} \) |
| 79 | \( 1 - 16T + 79T^{2} \) |
| 83 | \( 1 - 17.8T + 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 + 97T^{2} \) |
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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
−12.24585028390759671322413107395, −11.74690789160613403410368998186, −10.65259519130546393506529691558, −9.427181850360461378258420474759, −8.058574380337454448787468535229, −6.76301274587898694338444462086, −5.88368349574315693264548254986, −4.75691278882402142163935293995, −3.76993668551353217309528527282, −2.39073745840140963186525318063,
2.39073745840140963186525318063, 3.76993668551353217309528527282, 4.75691278882402142163935293995, 5.88368349574315693264548254986, 6.76301274587898694338444462086, 8.058574380337454448787468535229, 9.427181850360461378258420474759, 10.65259519130546393506529691558, 11.74690789160613403410368998186, 12.24585028390759671322413107395
