| L(s) = 1 | − 2·5-s − 6·9-s + 8·11-s − 4·13-s + 4·17-s − 2·19-s + 3·25-s − 4·29-s − 4·37-s + 4·41-s + 12·45-s + 16·47-s − 6·49-s − 20·53-s − 16·55-s − 8·59-s − 12·61-s + 8·65-s + 20·73-s + 27·81-s − 16·83-s − 8·85-s − 12·89-s + 4·95-s + 20·97-s − 48·99-s − 28·101-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 2·9-s + 2.41·11-s − 1.10·13-s + 0.970·17-s − 0.458·19-s + 3/5·25-s − 0.742·29-s − 0.657·37-s + 0.624·41-s + 1.78·45-s + 2.33·47-s − 6/7·49-s − 2.74·53-s − 2.15·55-s − 1.04·59-s − 1.53·61-s + 0.992·65-s + 2.34·73-s + 3·81-s − 1.75·83-s − 0.867·85-s − 1.27·89-s + 0.410·95-s + 2.03·97-s − 4.82·99-s − 2.78·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 36966400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36966400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 2 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 19 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 3 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 13 | $D_{4}$ | \( 1 + 4 T + 22 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_4$ | \( 1 - 4 T + 6 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 4 T + 30 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 4 T + 6 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 4 T + 54 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 16 T + 150 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 20 T + 198 T^{2} + 20 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 8 T + 102 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 110 T^{2} + p^{2} T^{4} \) |
| 73 | $C_4$ | \( 1 - 20 T + 214 T^{2} - 20 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 126 T^{2} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 16 T + 222 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 12 T + 182 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 20 T + 286 T^{2} - 20 p T^{3} + p^{2} T^{4} \) |
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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
−7.79326041691451744824114874533, −7.74269189016468895191506103051, −7.05982139281750810431961040207, −6.98216089460379869753799510756, −6.32984141942841291520839643160, −6.24404765592405336953248071773, −5.76066753189190162578981915101, −5.50433206236955424146553812031, −4.84102475910043083569448590987, −4.76652703118763336073145915062, −3.99062527219761074078557852225, −3.96388883699780316487747790824, −3.38254625659879094326083919654, −3.12753437923438136331575289245, −2.63197109039988661671267546503, −2.19766254208648998097436500755, −1.40155146488606841420170709823, −1.13699710816821334803465278597, 0, 0,
1.13699710816821334803465278597, 1.40155146488606841420170709823, 2.19766254208648998097436500755, 2.63197109039988661671267546503, 3.12753437923438136331575289245, 3.38254625659879094326083919654, 3.96388883699780316487747790824, 3.99062527219761074078557852225, 4.76652703118763336073145915062, 4.84102475910043083569448590987, 5.50433206236955424146553812031, 5.76066753189190162578981915101, 6.24404765592405336953248071773, 6.32984141942841291520839643160, 6.98216089460379869753799510756, 7.05982139281750810431961040207, 7.74269189016468895191506103051, 7.79326041691451744824114874533
