| L(s) = 1 | + 8·5-s − 4·13-s + 48·17-s − 2·25-s + 88·29-s + 36·37-s + 16·41-s + 82·49-s − 72·53-s − 124·61-s − 32·65-s − 132·73-s + 384·85-s − 288·89-s + 188·97-s − 40·101-s − 132·109-s − 160·113-s − 14·121-s − 344·125-s + 127-s + 131-s + 137-s + 139-s + 704·145-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | + 8/5·5-s − 0.307·13-s + 2.82·17-s − 0.0799·25-s + 3.03·29-s + 0.972·37-s + 0.390·41-s + 1.67·49-s − 1.35·53-s − 2.03·61-s − 0.492·65-s − 1.80·73-s + 4.51·85-s − 3.23·89-s + 1.93·97-s − 0.396·101-s − 1.21·109-s − 1.41·113-s − 0.115·121-s − 2.75·125-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 4.85·145-s + 0.00671·149-s + 0.00662·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 82944 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 82944 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(3.360621440\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.360621440\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| good | 5 | $C_2$ | \( ( 1 - 4 T + p^{2} T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 - 82 T^{2} + p^{4} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 14 T^{2} + p^{4} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p^{2} T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 24 T + p^{2} T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 146 T^{2} + p^{4} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 34 T^{2} + p^{4} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 44 T + p^{2} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 782 T^{2} + p^{4} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 18 T + p^{2} T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 8 T + p^{2} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 562 T^{2} + p^{4} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 3394 T^{2} + p^{4} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 36 T + p^{2} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 5938 T^{2} + p^{4} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 62 T + p^{2} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 2578 T^{2} + p^{4} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 6302 T^{2} + p^{4} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 66 T + p^{2} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 12082 T^{2} + p^{4} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 13522 T^{2} + p^{4} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 144 T + p^{2} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 94 T + p^{2} T^{2} )^{2} \) |
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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
−11.97786537610368831744611724368, −11.43247929634913797371129277588, −10.58841218467529952332109199689, −10.32941988415354615033434846519, −9.847593036175266977488611450277, −9.737310265691370571533512062388, −9.122703997262282740440070828707, −8.515857412531495271565344580034, −7.76481719397146669134277054123, −7.73318497556932393698796326015, −6.80282878682014131492312304220, −6.25629214995440026389079997125, −5.67810316514910430123091520950, −5.60222148218164875676868192122, −4.73899776071186295991942874121, −4.13251734496173816079442426557, −2.98034827262326603156441859644, −2.80970154986856513105290862797, −1.66044616267927121550268339006, −1.01753006214348937234375582991,
1.01753006214348937234375582991, 1.66044616267927121550268339006, 2.80970154986856513105290862797, 2.98034827262326603156441859644, 4.13251734496173816079442426557, 4.73899776071186295991942874121, 5.60222148218164875676868192122, 5.67810316514910430123091520950, 6.25629214995440026389079997125, 6.80282878682014131492312304220, 7.73318497556932393698796326015, 7.76481719397146669134277054123, 8.515857412531495271565344580034, 9.122703997262282740440070828707, 9.737310265691370571533512062388, 9.847593036175266977488611450277, 10.32941988415354615033434846519, 10.58841218467529952332109199689, 11.43247929634913797371129277588, 11.97786537610368831744611724368
