| L(s) = 1 | − 1.98e3·4-s − 1.27e6·11-s − 2.42e5·16-s + 3.90e7·19-s + 2.15e7·29-s − 1.01e8·31-s − 1.79e9·41-s + 2.53e9·44-s + 3.18e9·49-s + 1.11e9·59-s + 9.90e9·61-s + 8.81e9·64-s + 2.96e10·71-s − 7.75e10·76-s − 7.44e9·79-s − 5.09e10·89-s − 1.86e10·101-s + 1.38e11·109-s − 4.27e10·116-s + 6.49e11·121-s + 2.02e11·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | − 0.970·4-s − 2.38·11-s − 0.0577·16-s + 3.61·19-s + 0.194·29-s − 0.639·31-s − 2.42·41-s + 2.31·44-s + 1.61·49-s + 0.202·59-s + 1.50·61-s + 1.02·64-s + 1.95·71-s − 3.50·76-s − 0.272·79-s − 0.967·89-s − 0.176·101-s + 0.864·109-s − 0.188·116-s + 2.27·121-s + 0.620·124-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50625 ^{s/2} \, \Gamma_{\C}(s+11/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(2.095896800\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.095896800\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 + 497 p^{2} T^{2} + p^{22} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 3184035886 T^{2} + p^{22} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 637836 T + p^{11} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 2997236894278 T^{2} + p^{22} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 59030553017950 T^{2} + p^{22} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 1026916 p T + p^{11} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 1671151147058254 T^{2} + p^{22} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 10751262 T + p^{11} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 50937400 T + p^{11} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 86045127510168074 T^{2} + p^{22} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 898833450 T + p^{11} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 940924663945338070 T^{2} + p^{22} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 2523987300737098270 T^{2} + p^{22} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 4155644929310362294 T^{2} + p^{22} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 555306924 T + p^{11} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 4950420998 T + p^{11} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - \)\(21\!\cdots\!10\)\( T^{2} + p^{22} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 14831086248 T + p^{11} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - \)\(43\!\cdots\!54\)\( T^{2} + p^{22} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 3720542360 T + p^{11} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - \)\(24\!\cdots\!38\)\( T^{2} + p^{22} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 25472769174 T + p^{11} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - \)\(12\!\cdots\!90\)\( T^{2} + p^{22} T^{4} \) |
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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
−10.26013055624391579397031538763, −10.09015529269758679007322305970, −9.555387591150854491228361193190, −9.211363202809962890509834046899, −8.407291893574030246419461542305, −8.224083404088624410608608558863, −7.54687649002386147609834940059, −7.27171361506172162531113448445, −6.72523262772852294377239679544, −5.51692285230561765382721906970, −5.45132601243723580171335831832, −5.15697007072541759654204208499, −4.61643147270408097737706560964, −3.72594479081787418461783245612, −3.30920773648356184989822698093, −2.78842903682447883810081312813, −2.21808600646959205945183614418, −1.41755280980498497957603311656, −0.63159450375377273393932773780, −0.44909225289707802898581282973,
0.44909225289707802898581282973, 0.63159450375377273393932773780, 1.41755280980498497957603311656, 2.21808600646959205945183614418, 2.78842903682447883810081312813, 3.30920773648356184989822698093, 3.72594479081787418461783245612, 4.61643147270408097737706560964, 5.15697007072541759654204208499, 5.45132601243723580171335831832, 5.51692285230561765382721906970, 6.72523262772852294377239679544, 7.27171361506172162531113448445, 7.54687649002386147609834940059, 8.224083404088624410608608558863, 8.407291893574030246419461542305, 9.211363202809962890509834046899, 9.555387591150854491228361193190, 10.09015529269758679007322305970, 10.26013055624391579397031538763
