| L(s) = 1 | − 1.61·2-s + 0.618·4-s − 0.618·7-s + 2.23·8-s + 5.23·11-s − 1.85·13-s + 1.00·14-s − 4.85·16-s − 5.23·17-s + 0.854·19-s − 8.47·22-s + 3.76·23-s + 3·26-s − 0.381·28-s + 3.61·29-s − 3·31-s + 3.38·32-s + 8.47·34-s + 0.236·37-s − 1.38·38-s + 0.763·41-s + 4.85·43-s + 3.23·44-s − 6.09·46-s + 0.618·47-s − 6.61·49-s − 1.14·52-s + ⋯ |
| L(s) = 1 | − 1.14·2-s + 0.309·4-s − 0.233·7-s + 0.790·8-s + 1.57·11-s − 0.514·13-s + 0.267·14-s − 1.21·16-s − 1.26·17-s + 0.195·19-s − 1.80·22-s + 0.784·23-s + 0.588·26-s − 0.0721·28-s + 0.671·29-s − 0.538·31-s + 0.597·32-s + 1.45·34-s + 0.0388·37-s − 0.224·38-s + 0.119·41-s + 0.740·43-s + 0.487·44-s − 0.897·46-s + 0.0901·47-s − 0.945·49-s − 0.158·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9186100964\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9186100964\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 1.61T + 2T^{2} \) |
| 7 | \( 1 + 0.618T + 7T^{2} \) |
| 11 | \( 1 - 5.23T + 11T^{2} \) |
| 13 | \( 1 + 1.85T + 13T^{2} \) |
| 17 | \( 1 + 5.23T + 17T^{2} \) |
| 19 | \( 1 - 0.854T + 19T^{2} \) |
| 23 | \( 1 - 3.76T + 23T^{2} \) |
| 29 | \( 1 - 3.61T + 29T^{2} \) |
| 31 | \( 1 + 3T + 31T^{2} \) |
| 37 | \( 1 - 0.236T + 37T^{2} \) |
| 41 | \( 1 - 0.763T + 41T^{2} \) |
| 43 | \( 1 - 4.85T + 43T^{2} \) |
| 47 | \( 1 - 0.618T + 47T^{2} \) |
| 53 | \( 1 + 3.47T + 53T^{2} \) |
| 59 | \( 1 - 10.8T + 59T^{2} \) |
| 61 | \( 1 - 8.70T + 61T^{2} \) |
| 67 | \( 1 + 4.76T + 67T^{2} \) |
| 71 | \( 1 - 6.61T + 71T^{2} \) |
| 73 | \( 1 - 9T + 73T^{2} \) |
| 79 | \( 1 + 8.09T + 79T^{2} \) |
| 83 | \( 1 + 6.23T + 83T^{2} \) |
| 89 | \( 1 - 8.94T + 89T^{2} \) |
| 97 | \( 1 - 3.85T + 97T^{2} \) |
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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
−8.386857790819737592543259164109, −7.45433871200416759575763206496, −6.84080476266348659784289720818, −6.38803930995554257227690153276, −5.19099042436031083747374273038, −4.41992206897343276297631577842, −3.74371314898347007656658835269, −2.53404992054090660418478925006, −1.58594752228707010579155536978, −0.64015275388874822753373934600,
0.64015275388874822753373934600, 1.58594752228707010579155536978, 2.53404992054090660418478925006, 3.74371314898347007656658835269, 4.41992206897343276297631577842, 5.19099042436031083747374273038, 6.38803930995554257227690153276, 6.84080476266348659784289720818, 7.45433871200416759575763206496, 8.386857790819737592543259164109
