L(s) = 1 | − 18·11-s − 90·17-s − 106·19-s − 125·25-s + 522·41-s + 290·43-s − 343·49-s + 846·59-s + 70·67-s + 430·73-s − 1.35e3·83-s + 1.02e3·89-s − 1.91e3·97-s + 1.71e3·107-s + 270·113-s + ⋯ |
L(s) = 1 | − 0.493·11-s − 1.28·17-s − 1.27·19-s − 25-s + 1.98·41-s + 1.02·43-s − 49-s + 1.86·59-s + 0.127·67-s + 0.689·73-s − 1.78·83-s + 1.22·89-s − 1.99·97-s + 1.54·107-s + 0.224·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.384618123\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.384618123\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + p^{3} T^{2} \) |
| 7 | \( 1 + p^{3} T^{2} \) |
| 11 | \( 1 + 18 T + p^{3} T^{2} \) |
| 13 | \( 1 + p^{3} T^{2} \) |
| 17 | \( 1 + 90 T + p^{3} T^{2} \) |
| 19 | \( 1 + 106 T + p^{3} T^{2} \) |
| 23 | \( 1 + p^{3} T^{2} \) |
| 29 | \( 1 + p^{3} T^{2} \) |
| 31 | \( 1 + p^{3} T^{2} \) |
| 37 | \( 1 + p^{3} T^{2} \) |
| 41 | \( 1 - 522 T + p^{3} T^{2} \) |
| 43 | \( 1 - 290 T + p^{3} T^{2} \) |
| 47 | \( 1 + p^{3} T^{2} \) |
| 53 | \( 1 + p^{3} T^{2} \) |
| 59 | \( 1 - 846 T + p^{3} T^{2} \) |
| 61 | \( 1 + p^{3} T^{2} \) |
| 67 | \( 1 - 70 T + p^{3} T^{2} \) |
| 71 | \( 1 + p^{3} T^{2} \) |
| 73 | \( 1 - 430 T + p^{3} T^{2} \) |
| 79 | \( 1 + p^{3} T^{2} \) |
| 83 | \( 1 + 1350 T + p^{3} T^{2} \) |
| 89 | \( 1 - 1026 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1910 T + p^{3} T^{2} \) |
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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
−8.619281168638795112407886873390, −7.954978688110484957168866425106, −7.09934721546626694265244891312, −6.30691784901320170140524070220, −5.59184352983333613102173582038, −4.51857309700336600843735852499, −3.95407177813589944570145625268, −2.64647192880426427365007076132, −1.96420513759617558267845070780, −0.50346879801864770263529584095,
0.50346879801864770263529584095, 1.96420513759617558267845070780, 2.64647192880426427365007076132, 3.95407177813589944570145625268, 4.51857309700336600843735852499, 5.59184352983333613102173582038, 6.30691784901320170140524070220, 7.09934721546626694265244891312, 7.954978688110484957168866425106, 8.619281168638795112407886873390