| L(s) = 1 | − 2·2-s − 2·3-s − 4-s − 2·5-s + 4·6-s − 4·7-s + 8·8-s + 2·9-s + 4·10-s + 9·11-s + 2·12-s − 4·13-s + 8·14-s + 4·15-s − 7·16-s + 6·17-s − 4·18-s − 6·19-s + 2·20-s + 8·21-s − 18·22-s − 10·23-s − 16·24-s − 2·25-s + 8·26-s − 6·27-s + 4·28-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 1.15·3-s − 1/2·4-s − 0.894·5-s + 1.63·6-s − 1.51·7-s + 2.82·8-s + 2/3·9-s + 1.26·10-s + 2.71·11-s + 0.577·12-s − 1.10·13-s + 2.13·14-s + 1.03·15-s − 7/4·16-s + 1.45·17-s − 0.942·18-s − 1.37·19-s + 0.447·20-s + 1.74·21-s − 3.83·22-s − 2.08·23-s − 3.26·24-s − 2/5·25-s + 1.56·26-s − 1.15·27-s + 0.755·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17247409 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17247409 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 4153 | | \( 1+O(T) \) |
| good | 2 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 3 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 5 | $D_{4}$ | \( 1 + 2 T + 6 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 11 | $C_4$ | \( 1 - 9 T + 41 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 4 T + 10 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 6 T + 38 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 6 T + 42 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 10 T + 66 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 15 T + 113 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_4$ | \( 1 - 11 T + 91 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 3 T + 65 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 10 T + 62 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 18 T + 162 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 14 T + 98 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 18 T + 182 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 8 T + 54 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 9 T + 81 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 11 T + 103 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 8 T + 78 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 3 T + 137 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 146 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 97 | $D_{4}$ | \( 1 - 9 T + 213 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
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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
−8.197065737107628653214581694801, −8.040537210236257156354080220290, −7.64182546059456399306023814805, −7.03802760486957855221604446980, −6.95131427499005716899619982051, −6.40444239277073619809845806943, −6.07348155661884584221161914717, −5.97276281622027260997818792816, −5.20591580280943424651192111390, −4.72293403501254111141346206840, −4.31874520013031282198875423118, −4.13130446540562884816015177230, −3.80093699636833456010533041765, −3.43138457214038625736715256157, −2.60536762884842155831242671935, −1.89097121068751934730213482505, −1.04748893731106981159505805916, −1.00346494665051218367891109961, 0, 0,
1.00346494665051218367891109961, 1.04748893731106981159505805916, 1.89097121068751934730213482505, 2.60536762884842155831242671935, 3.43138457214038625736715256157, 3.80093699636833456010533041765, 4.13130446540562884816015177230, 4.31874520013031282198875423118, 4.72293403501254111141346206840, 5.20591580280943424651192111390, 5.97276281622027260997818792816, 6.07348155661884584221161914717, 6.40444239277073619809845806943, 6.95131427499005716899619982051, 7.03802760486957855221604446980, 7.64182546059456399306023814805, 8.040537210236257156354080220290, 8.197065737107628653214581694801
