| L(s) = 1 | − 6·3-s + 27·9-s − 26·13-s + 36·17-s − 48·23-s + 202·25-s − 108·27-s + 12·29-s + 156·39-s − 40·43-s + 578·49-s − 216·51-s − 612·53-s + 140·61-s + 288·69-s − 1.21e3·75-s + 832·79-s + 405·81-s − 72·87-s + 2.55e3·101-s + 1.75e3·107-s + 1.38e3·113-s − 702·117-s + 2.36e3·121-s + 127-s + 240·129-s + 131-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 9-s − 0.554·13-s + 0.513·17-s − 0.435·23-s + 1.61·25-s − 0.769·27-s + 0.0768·29-s + 0.640·39-s − 0.141·43-s + 1.68·49-s − 0.593·51-s − 1.58·53-s + 0.293·61-s + 0.502·69-s − 1.86·75-s + 1.18·79-s + 5/9·81-s − 0.0887·87-s + 2.51·101-s + 1.58·107-s + 1.14·113-s − 0.554·117-s + 1.77·121-s + 0.000698·127-s + 0.163·129-s + 0.000666·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 389376 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 389376 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.566220146\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.566220146\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_1$ | \( ( 1 + p T )^{2} \) |
| 13 | $C_2$ | \( 1 + 2 p T + p^{3} T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 202 T^{2} + p^{6} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 578 T^{2} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 2362 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 18 T + p^{3} T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 11690 T^{2} + p^{6} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 24 T + p^{3} T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 50 p^{2} T^{2} + p^{6} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 66314 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 125554 T^{2} + p^{6} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 20 T + p^{3} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 120946 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 306 T + p^{3} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 83750 T^{2} + p^{6} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 70 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 395594 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 342686 T^{2} + p^{6} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 777842 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 416 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 17078 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 1409170 T^{2} + p^{6} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 1548098 T^{2} + p^{6} T^{4} \) |
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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
−10.49361312444175652880191203906, −10.08233209661012421080280868651, −9.691089477507971603465867743613, −9.204462708357827299556523815611, −8.686421092017824869469162151228, −8.277717279890582151086819338876, −7.48944476379671863725420031909, −7.41149842440090257512723258767, −6.81854566334900337736092477414, −6.26354100113121662877903392870, −5.99428538010790908656445017488, −5.37400025855853906992394171624, −4.81767351197160746903693914730, −4.69823802741809431962518271939, −3.85566403046489682942511542100, −3.31735195520040350489574850926, −2.57538884598504334606969270529, −1.85941798453617844214838396428, −1.02127159714424799426160220637, −0.47359552811597610726077333166,
0.47359552811597610726077333166, 1.02127159714424799426160220637, 1.85941798453617844214838396428, 2.57538884598504334606969270529, 3.31735195520040350489574850926, 3.85566403046489682942511542100, 4.69823802741809431962518271939, 4.81767351197160746903693914730, 5.37400025855853906992394171624, 5.99428538010790908656445017488, 6.26354100113121662877903392870, 6.81854566334900337736092477414, 7.41149842440090257512723258767, 7.48944476379671863725420031909, 8.277717279890582151086819338876, 8.686421092017824869469162151228, 9.204462708357827299556523815611, 9.691089477507971603465867743613, 10.08233209661012421080280868651, 10.49361312444175652880191203906
