| L(s) = 1 | − 2-s − 4·3-s − 2·4-s + 5-s + 4·6-s − 2·7-s + 3·8-s + 6·9-s − 10-s + 6·11-s + 8·12-s − 7·13-s + 2·14-s − 4·15-s + 16-s + 6·17-s − 6·18-s − 19-s − 2·20-s + 8·21-s − 6·22-s − 12·24-s − 8·25-s + 7·26-s + 4·27-s + 4·28-s + 13·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 2.30·3-s − 4-s + 0.447·5-s + 1.63·6-s − 0.755·7-s + 1.06·8-s + 2·9-s − 0.316·10-s + 1.80·11-s + 2.30·12-s − 1.94·13-s + 0.534·14-s − 1.03·15-s + 1/4·16-s + 1.45·17-s − 1.41·18-s − 0.229·19-s − 0.447·20-s + 1.74·21-s − 1.27·22-s − 2.44·24-s − 8/5·25-s + 1.37·26-s + 0.769·27-s + 0.755·28-s + 2.41·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10426441 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10426441 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3229 | | \( 1+O(T) \) |
| good | 2 | $D_{4}$ | \( 1 + T + 3 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 3 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 5 | $D_{4}$ | \( 1 - T + 9 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 11 | $C_4$ | \( 1 - 6 T + 26 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 7 T + 37 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 6 T + 38 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + T + 27 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 13 T + 99 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 17 T + 133 T^{2} + 17 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 5 T + 49 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_4$ | \( 1 - 9 T + 71 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + T + 55 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 89 T^{2} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 6 T + 95 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 113 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 67 | $D_{4}$ | \( 1 - 12 T + 125 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 12 T + 133 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 2 T - 33 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 18 T + 194 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 15 T + 221 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 9 T + 47 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + T + 183 T^{2} + p T^{3} + p^{2} T^{4} \) |
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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
−8.427060992394067699502411002199, −8.261546788582151084028290776978, −7.49468130967194770377486496147, −7.45416931600273716256954142346, −6.83204308362645302425423089947, −6.35112862936398126803143441163, −6.30265680586406888510255250824, −5.87566955879507125629408954491, −5.33807319002641422235476334901, −5.22911614526359833785589068610, −4.64707822353024275204256149209, −4.53639698697072475497395211070, −3.67416701633506898483955354808, −3.63285882790870401122413794115, −2.71641034710618162946199146929, −2.14759954733567702445904590174, −1.20899968350684411097755617824, −0.959966070508571975900328113569, 0, 0,
0.959966070508571975900328113569, 1.20899968350684411097755617824, 2.14759954733567702445904590174, 2.71641034710618162946199146929, 3.63285882790870401122413794115, 3.67416701633506898483955354808, 4.53639698697072475497395211070, 4.64707822353024275204256149209, 5.22911614526359833785589068610, 5.33807319002641422235476334901, 5.87566955879507125629408954491, 6.30265680586406888510255250824, 6.35112862936398126803143441163, 6.83204308362645302425423089947, 7.45416931600273716256954142346, 7.49468130967194770377486496147, 8.261546788582151084028290776978, 8.427060992394067699502411002199
