| L(s) = 1 | + 12·7-s + 50·9-s + 52·17-s − 156·23-s − 25·25-s + 216·31-s − 44·41-s + 1.02e3·47-s − 578·49-s + 600·63-s + 824·71-s + 1.75e3·73-s − 1.20e3·79-s + 1.77e3·81-s + 300·89-s + 772·97-s − 1.19e3·103-s + 3.12e3·113-s + 624·119-s + 1.63e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 2.60e3·153-s + ⋯ |
| L(s) = 1 | + 0.647·7-s + 1.85·9-s + 0.741·17-s − 1.41·23-s − 1/5·25-s + 1.25·31-s − 0.167·41-s + 3.19·47-s − 1.68·49-s + 1.19·63-s + 1.37·71-s + 2.81·73-s − 1.70·79-s + 2.42·81-s + 0.357·89-s + 0.808·97-s − 1.14·103-s + 2.60·113-s + 0.480·119-s + 1.23·121-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 + 1.37·153-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638400 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(5.429300315\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.429300315\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + p^{2} T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 50 T^{2} + p^{6} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 6 T + p^{3} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 1638 T^{2} + p^{6} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 2950 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 26 T + p^{3} T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 3718 T^{2} + p^{6} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 78 T + p^{3} T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 46278 T^{2} + p^{6} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 108 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 30550 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 22 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 36350 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 514 T + p^{3} T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 297750 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 160758 T^{2} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 185638 T^{2} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 585650 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 412 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 878 T + p^{3} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 600 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 1064050 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 150 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 386 T + p^{3} T^{2} )^{2} \) |
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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
−9.460919237367987386630955219353, −9.353091576338153777065040277663, −8.558136658489028515859395247561, −8.134781940972723381406904005667, −7.941573185693595096930513904322, −7.53051362701672748137370662557, −6.93232800126376933627784108416, −6.87271300696606037733046787266, −5.99169462256673363823139598580, −5.96866810952229339385441373898, −5.11505881291799593144907487573, −4.87625591156537573905728024493, −4.19840839434570197100600848614, −4.08796846361579031714144031659, −3.50406832653571761706886127628, −2.81516351405090479808850808159, −1.97159307984987243176442238533, −1.87038034617690268308018810714, −0.988476205509341650252960853419, −0.65157444349750756472724535997,
0.65157444349750756472724535997, 0.988476205509341650252960853419, 1.87038034617690268308018810714, 1.97159307984987243176442238533, 2.81516351405090479808850808159, 3.50406832653571761706886127628, 4.08796846361579031714144031659, 4.19840839434570197100600848614, 4.87625591156537573905728024493, 5.11505881291799593144907487573, 5.96866810952229339385441373898, 5.99169462256673363823139598580, 6.87271300696606037733046787266, 6.93232800126376933627784108416, 7.53051362701672748137370662557, 7.941573185693595096930513904322, 8.134781940972723381406904005667, 8.558136658489028515859395247561, 9.353091576338153777065040277663, 9.460919237367987386630955219353
