| L(s) = 1 | + 6·3-s + 27·9-s + 28·17-s − 144·23-s + 246·25-s + 108·27-s + 204·29-s + 280·43-s − 338·49-s + 168·51-s + 1.05e3·53-s − 820·61-s − 864·69-s + 1.47e3·75-s − 1.28e3·79-s + 405·81-s + 1.22e3·87-s + 1.22e3·101-s + 2.70e3·103-s − 3.03e3·113-s − 1.96e3·121-s + 127-s + 1.68e3·129-s + 131-s + 137-s + 139-s − 2.02e3·147-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 9-s + 0.399·17-s − 1.30·23-s + 1.96·25-s + 0.769·27-s + 1.30·29-s + 0.993·43-s − 0.985·49-s + 0.461·51-s + 2.72·53-s − 1.72·61-s − 1.50·69-s + 2.27·75-s − 1.82·79-s + 5/9·81-s + 1.50·87-s + 1.20·101-s + 2.58·103-s − 2.52·113-s − 1.47·121-s + 0.000698·127-s + 1.14·129-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s − 1.13·147-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4112784 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4112784 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(5.324911551\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.324911551\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - 246 T^{2} + p^{6} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 338 T^{2} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 1962 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 14 T + p^{3} T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 13702 T^{2} + p^{6} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 72 T + p^{3} T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 102 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 41086 T^{2} + p^{6} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 47690 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 75342 T^{2} + p^{6} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 140 T + p^{3} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 120030 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 526 T + p^{3} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 300534 T^{2} + p^{6} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 410 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 246310 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 58578 T^{2} + p^{6} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 521998 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 640 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 760826 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 692562 T^{2} + p^{6} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 1626430 T^{2} + p^{6} T^{4} \) |
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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
−9.124274325822656895203433760919, −8.519926980992739635935754671017, −8.374353947812951283536540023160, −7.71823997685475186750354013703, −7.64628119322417793580509004451, −7.06620956274989615360234939962, −6.75685636226184684741923616383, −6.22415003086038462786123877335, −5.96066261953728101922559712026, −5.23672851518401069917398648869, −4.94688989206704124506213762350, −4.39768572170333406033630841599, −4.02445682722962573778310429542, −3.60408325007648737536345349268, −2.99794688425899016482321795487, −2.64924450538313261881176546796, −2.33425926609340014814110981864, −1.48225111171507079653476834222, −1.16898147348237885581706773418, −0.44891388009551541113233175732,
0.44891388009551541113233175732, 1.16898147348237885581706773418, 1.48225111171507079653476834222, 2.33425926609340014814110981864, 2.64924450538313261881176546796, 2.99794688425899016482321795487, 3.60408325007648737536345349268, 4.02445682722962573778310429542, 4.39768572170333406033630841599, 4.94688989206704124506213762350, 5.23672851518401069917398648869, 5.96066261953728101922559712026, 6.22415003086038462786123877335, 6.75685636226184684741923616383, 7.06620956274989615360234939962, 7.64628119322417793580509004451, 7.71823997685475186750354013703, 8.374353947812951283536540023160, 8.519926980992739635935754671017, 9.124274325822656895203433760919
