| L(s) = 1 | − 4·4-s − 60·11-s + 16·16-s + 278·19-s + 468·29-s − 110·31-s − 276·41-s + 240·44-s + 157·49-s − 792·59-s + 46·61-s − 64·64-s − 408·71-s − 1.11e3·76-s + 1.41e3·79-s − 1.63e3·89-s + 2.55e3·101-s − 1.18e3·109-s − 1.87e3·116-s + 38·121-s + 440·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 1.64·11-s + 1/4·16-s + 3.35·19-s + 2.99·29-s − 0.637·31-s − 1.05·41-s + 0.822·44-s + 0.457·49-s − 1.74·59-s + 0.0965·61-s − 1/8·64-s − 0.681·71-s − 1.67·76-s + 2.01·79-s − 1.94·89-s + 2.51·101-s − 1.04·109-s − 1.49·116-s + 0.0285·121-s + 0.318·124-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1822500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1822500 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.726725597\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.726725597\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( 1 + p^{2} T^{2} \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( 1 - 157 T^{2} + p^{6} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 30 T + p^{3} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 3238 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 8062 T^{2} + p^{6} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 139 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 12530 T^{2} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 234 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 55 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 64825 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 138 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 156205 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 73690 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 188854 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 396 T + p^{3} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 23 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 397222 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 204 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 300553 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 709 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 62030 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 816 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 1006321 T^{2} + p^{6} T^{4} \) |
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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
−9.570481969006266341549154790372, −8.899284839362431612641363392741, −8.771931785847131266481168715539, −8.099994957858533972928547740370, −7.85234797402414427356938183499, −7.41386820279147379520463063780, −7.27971347753815943053322472452, −6.38842106188074351063281374189, −6.27217831048045250812603944092, −5.42332533482603929568606379003, −5.09885397962351198421637039525, −5.09589496011161255403879044032, −4.49594481256054279876016463412, −3.69487395534410949852192498425, −3.32344450262716602473195687788, −2.69162723651707337127576603747, −2.63738695923539061970066372381, −1.40956869452361970055702115564, −1.09067387698420127719921167803, −0.34720063351703989519013264430,
0.34720063351703989519013264430, 1.09067387698420127719921167803, 1.40956869452361970055702115564, 2.63738695923539061970066372381, 2.69162723651707337127576603747, 3.32344450262716602473195687788, 3.69487395534410949852192498425, 4.49594481256054279876016463412, 5.09589496011161255403879044032, 5.09885397962351198421637039525, 5.42332533482603929568606379003, 6.27217831048045250812603944092, 6.38842106188074351063281374189, 7.27971347753815943053322472452, 7.41386820279147379520463063780, 7.85234797402414427356938183499, 8.099994957858533972928547740370, 8.771931785847131266481168715539, 8.899284839362431612641363392741, 9.570481969006266341549154790372
