| L(s) = 1 | − 1.02e3·4-s − 1.90e5·9-s + 1.53e6·11-s + 1.04e6·16-s + 1.10e7·19-s + 3.05e7·29-s − 4.83e8·31-s + 1.94e8·36-s − 2.43e9·41-s − 1.57e9·44-s + 3.30e9·49-s + 2.13e9·59-s − 4.18e9·61-s − 1.07e9·64-s + 1.93e10·71-s − 1.13e10·76-s − 1.00e10·79-s + 4.85e9·81-s − 7.11e10·89-s − 2.92e11·99-s − 1.76e11·101-s + 1.93e11·109-s − 3.12e10·116-s + 1.20e12·121-s + 4.94e11·124-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 1.07·9-s + 2.87·11-s + 1/4·16-s + 1.02·19-s + 0.276·29-s − 3.03·31-s + 0.537·36-s − 3.28·41-s − 1.43·44-s + 1.66·49-s + 0.389·59-s − 0.634·61-s − 1/8·64-s + 1.27·71-s − 0.511·76-s − 0.367·79-s + 0.154·81-s − 1.34·89-s − 3.09·99-s − 1.67·101-s + 1.20·109-s − 0.138·116-s + 4.22·121-s + 1.51·124-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2500 ^{s/2} \, \Gamma_{\C}(s+11/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(1.944816940\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.944816940\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_2$ | \( 1 + p^{10} T^{2} \) |
| 5 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 + 2350 p^{4} T^{2} + p^{22} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 3300624010 T^{2} + p^{22} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 769152 T + p^{11} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 2739792871750 T^{2} + p^{22} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 37809927471170 T^{2} + p^{22} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 5521660 T + p^{11} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 307745049437770 T^{2} + p^{22} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 15269010 T + p^{11} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 241583788 T + p^{11} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 355172106587829910 T^{2} + p^{22} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 1217700138 T + p^{11} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 1391502354725912770 T^{2} + p^{22} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 2580734140154678170 T^{2} + p^{22} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 5772519631420258390 T^{2} + p^{22} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 1069039020 T + p^{11} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 2091535078 T + p^{11} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - \)\(24\!\cdots\!70\)\( T^{2} + p^{22} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 9660178332 T + p^{11} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - \)\(59\!\cdots\!10\)\( T^{2} + p^{22} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 5026936280 T + p^{11} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - \)\(11\!\cdots\!90\)\( T^{2} + p^{22} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 35558583210 T + p^{11} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - \)\(14\!\cdots\!10\)\( T^{2} + p^{22} T^{4} \) |
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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
−13.82769493344400355724605535809, −12.84094791361212351450409990256, −12.02863045394149572243413709304, −11.85579964257498970280137093382, −11.28906801214359353324378329024, −10.59633224585332346227590844853, −9.577637996368804841097005670537, −9.316861489261240639280842434663, −8.747855133869243472275023092923, −8.273258273009780136152880398762, −7.02146431354943971697093230308, −6.88426667724267760994203351060, −5.79310703056785406080803712480, −5.42164230827264856408123896171, −4.37887344370982526321822840442, −3.59670398752803617786448697997, −3.32029807235470095507372730290, −1.89208478088729924246311446951, −1.32208407137078669735297116324, −0.43359200484065664203038252802,
0.43359200484065664203038252802, 1.32208407137078669735297116324, 1.89208478088729924246311446951, 3.32029807235470095507372730290, 3.59670398752803617786448697997, 4.37887344370982526321822840442, 5.42164230827264856408123896171, 5.79310703056785406080803712480, 6.88426667724267760994203351060, 7.02146431354943971697093230308, 8.273258273009780136152880398762, 8.747855133869243472275023092923, 9.316861489261240639280842434663, 9.577637996368804841097005670537, 10.59633224585332346227590844853, 11.28906801214359353324378329024, 11.85579964257498970280137093382, 12.02863045394149572243413709304, 12.84094791361212351450409990256, 13.82769493344400355724605535809
