| L(s) = 1 | − 1.98e3·4-s − 5.90e4·9-s + 1.27e6·11-s − 2.42e5·16-s + 3.90e7·19-s − 2.15e7·29-s − 1.01e8·31-s + 1.17e8·36-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 + 3.48e9·81-s + 5.09e10·89-s − 7.53e10·99-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 + ⋯ |
| L(s) = 1 | − 0.970·4-s − 1/3·9-s + 2.38·11-s − 0.0577·16-s + 3.61·19-s − 0.194·29-s − 0.639·31-s + 0.323·36-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 + 1/9·81-s + 0.967·89-s − 0.796·99-s + 0.176·101-s + 0.864·109-s + 0.188·116-s + 2.27·121-s + 0.620·124-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s+11/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(3.952388814\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.952388814\) |
| \(L(\frac{13}{2})\) |
|
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 | $C_2$ | \( 1 + p^{10} T^{2} \) |
| 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
−13.24597110925213487606870863041, −11.84185297309472189836348725867, −11.62276366668292594002410527669, −11.08702267711636584888750912268, −10.08264665092142166876438068054, −9.493968086180175223406914053097, −9.103409622778418405922850307344, −9.032461785825729146848806793371, −7.985398779837457761472434979246, −7.28439956589987458428025866658, −6.93244010083236649632234044173, −5.84165615633418547190690076638, −5.59207170534639445249193258633, −4.69645742058828334966016842719, −4.04156923760057073940327531615, −3.56215768470486954414004814293, −2.84990930028881598923641534069, −1.70448175178137811961052106873, −0.861731648869165037474324024716, −0.75955282088824200644808601092,
0.75955282088824200644808601092, 0.861731648869165037474324024716, 1.70448175178137811961052106873, 2.84990930028881598923641534069, 3.56215768470486954414004814293, 4.04156923760057073940327531615, 4.69645742058828334966016842719, 5.59207170534639445249193258633, 5.84165615633418547190690076638, 6.93244010083236649632234044173, 7.28439956589987458428025866658, 7.985398779837457761472434979246, 9.032461785825729146848806793371, 9.103409622778418405922850307344, 9.493968086180175223406914053097, 10.08264665092142166876438068054, 11.08702267711636584888750912268, 11.62276366668292594002410527669, 11.84185297309472189836348725867, 13.24597110925213487606870863041
