| L(s) = 1 | − 9·9-s − 32·11-s + 8·19-s − 164·29-s + 16·31-s − 492·41-s + 286·49-s − 1.18e3·59-s + 1.14e3·61-s − 1.53e3·71-s + 816·79-s + 81·81-s + 1.02e3·89-s + 288·99-s + 1.33e3·101-s − 4.15e3·109-s − 1.89e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 1.03e3·169-s + ⋯ |
| L(s) = 1 | − 1/3·9-s − 0.877·11-s + 0.0965·19-s − 1.05·29-s + 0.0926·31-s − 1.87·41-s + 0.833·49-s − 2.61·59-s + 2.40·61-s − 2.56·71-s + 1.16·79-s + 1/9·81-s + 1.21·89-s + 0.292·99-s + 1.31·101-s − 3.65·109-s − 1.42·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s + 0.468·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440000 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.08699678129\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.08699678129\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( 1 + p^{2} T^{2} \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( 1 - 286 T^{2} + p^{6} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 16 T + p^{3} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 1030 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 8382 T^{2} + p^{6} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 17934 T^{2} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 82 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 80170 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 6 p T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 115562 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 7650 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 195050 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 592 T + p^{3} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 574 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 571942 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 768 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 466670 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 408 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 1116678 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 510 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 1561150 T^{2} + p^{6} T^{4} \) |
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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
−10.02866430860871430518041470068, −9.090669181978706203389298612804, −8.876090310148763839061158426436, −8.207360814132493406067201718297, −8.075888973887839536725615142314, −7.38574212423257680935513288630, −7.35270528791872588849527886743, −6.55644521572146388225158567391, −6.34421325173085146227280498293, −5.66541438332756006412181027524, −5.40127581696242711437081352273, −4.91928112569723091465625334720, −4.52854576293546265263863441657, −3.64455583785418963353136530215, −3.61656439470327130558174019376, −2.73015135607053421510211272934, −2.45513548357055589123168139410, −1.69514718777245378304922472096, −1.13207584373531598017376621627, −0.07232626084317088969834298938,
0.07232626084317088969834298938, 1.13207584373531598017376621627, 1.69514718777245378304922472096, 2.45513548357055589123168139410, 2.73015135607053421510211272934, 3.61656439470327130558174019376, 3.64455583785418963353136530215, 4.52854576293546265263863441657, 4.91928112569723091465625334720, 5.40127581696242711437081352273, 5.66541438332756006412181027524, 6.34421325173085146227280498293, 6.55644521572146388225158567391, 7.35270528791872588849527886743, 7.38574212423257680935513288630, 8.075888973887839536725615142314, 8.207360814132493406067201718297, 8.876090310148763839061158426436, 9.090669181978706203389298612804, 10.02866430860871430518041470068
