| L(s) = 1 | + 11·7-s + 15·19-s − 25·25-s − 33·31-s + 47·37-s + 166·43-s + 72·49-s + 168·61-s − 109·67-s + 51·73-s + 131·79-s − 351·103-s + 143·109-s + 121·121-s + 127-s + 131-s + 165·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 337·169-s + 173-s − 275·175-s + ⋯ |
| L(s) = 1 | + 11/7·7-s + 0.789·19-s − 25-s − 1.06·31-s + 1.27·37-s + 3.86·43-s + 1.46·49-s + 2.75·61-s − 1.62·67-s + 0.698·73-s + 1.65·79-s − 3.40·103-s + 1.31·109-s + 121-s + 0.00787·127-s + 0.00763·131-s + 1.24·133-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s − 1.99·169-s + 0.00578·173-s − 1.57·175-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1016064 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1016064 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(3.873987229\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.873987229\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 7 | $C_2$ | \( 1 - 11 T + p^{2} T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - p^{2} T^{2} + p^{4} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - T + p^{2} T^{2} )( 1 + T + p^{2} T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 26 T + p^{2} T^{2} )( 1 + 11 T + p^{2} T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - p^{2} T^{2} + p^{4} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 13 T + p^{2} T^{2} )( 1 + 46 T + p^{2} T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 73 T + p^{2} T^{2} )( 1 + 26 T + p^{2} T^{2} ) \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 83 T + p^{2} T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 - p^{2} T^{2} + p^{4} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 121 T + p^{2} T^{2} )( 1 - 47 T + p^{2} T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 13 T + p^{2} T^{2} )( 1 + 122 T + p^{2} T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 97 T + p^{2} T^{2} )( 1 + 46 T + p^{2} T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 142 T + p^{2} T^{2} )( 1 + 11 T + p^{2} T^{2} ) \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 89 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p^{2} T^{2} )( 1 + 2 T + p^{2} T^{2} ) \) |
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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
−9.711684813060256571296528919157, −9.678700324405569018877628633601, −9.167911810484923979787621597143, −8.738386528741631877959795947785, −8.167394600741244506970976173251, −7.964806318823996140728831499573, −7.41992769751602685962718454440, −7.31210764455332505763592713756, −6.66044361692583330917486208986, −5.92673473001591665415986171910, −5.51541034061001099228545716879, −5.49065978869160734913704435982, −4.48435742174109074180335991185, −4.43813720504957652866900645661, −3.82677786692247108596845876722, −3.20561195909867726864755568519, −2.26400895768902160722413595289, −2.18375068163030849990210978951, −1.20140459710847769861807153748, −0.69889153736579179410188772967,
0.69889153736579179410188772967, 1.20140459710847769861807153748, 2.18375068163030849990210978951, 2.26400895768902160722413595289, 3.20561195909867726864755568519, 3.82677786692247108596845876722, 4.43813720504957652866900645661, 4.48435742174109074180335991185, 5.49065978869160734913704435982, 5.51541034061001099228545716879, 5.92673473001591665415986171910, 6.66044361692583330917486208986, 7.31210764455332505763592713756, 7.41992769751602685962718454440, 7.964806318823996140728831499573, 8.167394600741244506970976173251, 8.738386528741631877959795947785, 9.167911810484923979787621597143, 9.678700324405569018877628633601, 9.711684813060256571296528919157
