| L(s) = 1 | + 40·7-s + 50·9-s − 132·17-s + 344·23-s + 214·25-s + 256·31-s − 404·41-s + 816·47-s + 514·49-s + 2.00e3·63-s − 1.40e3·71-s + 836·73-s − 1.48e3·79-s + 1.77e3·81-s + 164·89-s − 2.24e3·97-s + 1.57e3·103-s − 1.08e3·113-s − 5.28e3·119-s + 2.46e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 6.60e3·153-s + ⋯ |
| L(s) = 1 | + 2.15·7-s + 1.85·9-s − 1.88·17-s + 3.11·23-s + 1.71·25-s + 1.48·31-s − 1.53·41-s + 2.53·47-s + 1.49·49-s + 3.99·63-s − 2.34·71-s + 1.34·73-s − 2.11·79-s + 2.42·81-s + 0.195·89-s − 2.34·97-s + 1.50·103-s − 0.902·113-s − 4.06·119-s + 1.85·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s − 3.48·153-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 65536 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65536 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(4.665288863\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.665288863\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 - 50 T^{2} + p^{6} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 214 T^{2} + p^{6} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 20 T + p^{3} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 2466 T^{2} + p^{6} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 1478 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 66 T + p^{3} T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 + 12526 T^{2} + p^{6} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 172 T + p^{3} T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 48774 T^{2} + p^{6} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 128 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 76342 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 202 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70210 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 408 T + p^{3} T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 + 178346 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 307074 T^{2} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 365158 T^{2} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 560722 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 700 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 418 T + p^{3} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 744 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 683890 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 82 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 1122 T + p^{3} T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.48733560445079692264879132999, −11.44219998288083194948232067544, −10.73170998342869394310116125406, −10.64310340627630115152013957873, −10.08000811755738108822100566793, −9.226727554482896655655391608134, −8.720876015295427429896313739246, −8.679971080444354889778193315343, −7.83929309060385999874776174053, −7.30431033963732812057856582035, −6.76153876800283296971972619656, −6.75486291685059564076216632424, −5.42770628692440714392831437724, −4.90063092845972977905959074812, −4.45574923018987764525705688769, −4.39276264398433548185009259496, −3.07170465024721629943019246653, −2.28258213412362568623907717066, −1.35949382730584233736380939030, −1.05356037827133218037755388089,
1.05356037827133218037755388089, 1.35949382730584233736380939030, 2.28258213412362568623907717066, 3.07170465024721629943019246653, 4.39276264398433548185009259496, 4.45574923018987764525705688769, 4.90063092845972977905959074812, 5.42770628692440714392831437724, 6.75486291685059564076216632424, 6.76153876800283296971972619656, 7.30431033963732812057856582035, 7.83929309060385999874776174053, 8.679971080444354889778193315343, 8.720876015295427429896313739246, 9.226727554482896655655391608134, 10.08000811755738108822100566793, 10.64310340627630115152013957873, 10.73170998342869394310116125406, 11.44219998288083194948232067544, 11.48733560445079692264879132999
