| L(s) = 1 | + 960·4-s + 2.24e5·11-s − 3.27e6·16-s + 2.74e7·19-s − 9.06e6·29-s − 5.30e8·31-s + 2.53e9·41-s + 2.15e8·44-s + 3.17e9·49-s + 1.49e10·59-s − 8.12e9·61-s − 7.16e9·64-s + 2.65e10·71-s + 2.63e10·76-s − 5.42e10·79-s − 1.27e11·89-s + 2.42e11·101-s − 4.65e11·109-s − 8.70e9·116-s − 5.32e11·121-s − 5.09e11·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | + 0.468·4-s + 0.419·11-s − 0.780·16-s + 2.54·19-s − 0.0820·29-s − 3.32·31-s + 3.41·41-s + 0.196·44-s + 1.60·49-s + 2.73·59-s − 1.23·61-s − 0.834·64-s + 1.74·71-s + 1.19·76-s − 1.98·79-s − 2.41·89-s + 2.29·101-s − 2.89·109-s − 0.0384·116-s − 1.86·121-s − 1.56·124-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
| \(L(6)\) |
\(\approx\) |
\(3.137804837\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.137804837\) |
| \(L(\frac{13}{2})\) |
|
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 | $C_2^2$ | \( 1 - 15 p^{6} T^{2} + p^{22} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 3171549230 T^{2} + p^{22} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 112028 T + p^{11} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 2381082913990 T^{2} + p^{22} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 68481509426910 T^{2} + p^{22} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 13712420 T + p^{11} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 192013219177870 T^{2} + p^{22} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 4533850 T + p^{11} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 265339008 T + p^{11} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 310833159700713910 T^{2} + p^{22} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 1266969958 T + p^{11} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 1858387834815761110 T^{2} + p^{22} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 2116783152048544290 T^{2} + p^{22} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 12765108038242661910 T^{2} + p^{22} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 7498737220 T + p^{11} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 4064828858 T + p^{11} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - \)\(19\!\cdots\!30\)\( T^{2} + p^{22} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 13283734648 T + p^{11} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + \)\(20\!\cdots\!90\)\( T^{2} + p^{22} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 27100302240 T + p^{11} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - \)\(13\!\cdots\!10\)\( T^{2} + p^{22} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 63500412630 T + p^{11} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - \)\(13\!\cdots\!50\)\( T^{2} + p^{22} T^{4} \) |
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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
−11.11329430320762930008724084840, −9.955303342770176691053875644329, −9.472103509086568639972299389200, −8.988333886200108515756155576483, −8.954247056437372749902772401486, −7.68936182106701708222229526101, −7.67057997452567959995970689210, −7.11514091461984694406547701932, −6.73846405349477300316683055529, −5.90373262393057302887335152059, −5.36065557163288725549745317087, −5.35586286003015381588630219070, −4.08477497754799349387988147649, −4.03547748874305910360292856685, −3.26371633874707613012763672615, −2.60068655070589100179076156227, −2.23718820049975134480606722816, −1.39687100343780032407590121781, −1.05813111363330051784176593555, −0.35260963102119723228072847195,
0.35260963102119723228072847195, 1.05813111363330051784176593555, 1.39687100343780032407590121781, 2.23718820049975134480606722816, 2.60068655070589100179076156227, 3.26371633874707613012763672615, 4.03547748874305910360292856685, 4.08477497754799349387988147649, 5.35586286003015381588630219070, 5.36065557163288725549745317087, 5.90373262393057302887335152059, 6.73846405349477300316683055529, 7.11514091461984694406547701932, 7.67057997452567959995970689210, 7.68936182106701708222229526101, 8.954247056437372749902772401486, 8.988333886200108515756155576483, 9.472103509086568639972299389200, 9.955303342770176691053875644329, 11.11329430320762930008724084840
