| L(s) = 1 | + 2·2-s + 4·3-s + 4-s + 2·5-s + 8·6-s + 6·9-s + 4·10-s − 2·11-s + 4·12-s + 8·13-s + 8·15-s + 16-s − 4·17-s + 12·18-s − 2·19-s + 2·20-s − 4·22-s + 8·23-s + 3·25-s + 16·26-s − 4·27-s + 12·29-s + 16·30-s + 8·31-s − 2·32-s − 8·33-s − 8·34-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 2.30·3-s + 1/2·4-s + 0.894·5-s + 3.26·6-s + 2·9-s + 1.26·10-s − 0.603·11-s + 1.15·12-s + 2.21·13-s + 2.06·15-s + 1/4·16-s − 0.970·17-s + 2.82·18-s − 0.458·19-s + 0.447·20-s − 0.852·22-s + 1.66·23-s + 3/5·25-s + 3.13·26-s − 0.769·27-s + 2.22·29-s + 2.92·30-s + 1.43·31-s − 0.353·32-s − 1.39·33-s − 1.37·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1092025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1092025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(11.90952534\) |
| \(L(\frac12)\) |
\(\approx\) |
\(11.90952534\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 11 | $C_1$ | \( ( 1 + T )^{2} \) |
| 19 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 2 | $D_{4}$ | \( 1 - p T + 3 T^{2} - p^{2} T^{3} + p^{2} T^{4} \) |
| 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 8 T + 34 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 4 T + 30 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 29 | $D_{4}$ | \( 1 - 12 T + 86 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 8 T + 70 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 8 T + 82 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 12 T + 110 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 8 T + 94 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 + 98 T^{2} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 8 T + 62 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 4 T + 94 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 12 T + 138 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 24 T + 278 T^{2} - 24 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 4 T + 78 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 158 T^{2} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 4 T + 54 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 8 T + 10 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
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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
−9.982464415011675720684641125250, −9.785512387266771973485987118759, −8.887186370007727515213713566823, −8.859645511320313339269777859808, −8.430605539601025991731967657355, −8.372993657359898213658539994643, −7.931134464477924215164870637458, −6.94037299505054646117398591462, −6.72867725196154039860846369462, −6.42265024675426580789242443153, −5.62101924871643653198916000849, −5.35185830673477984120739825870, −4.69467750720940404804277596242, −4.46150960180458677460823007145, −3.58569590515414638411341916070, −3.45343494455739933102218500763, −3.04104925521471203807922412918, −2.47803105698748227074678920147, −1.94942679495648786627647621890, −1.23022502596973767761813067277,
1.23022502596973767761813067277, 1.94942679495648786627647621890, 2.47803105698748227074678920147, 3.04104925521471203807922412918, 3.45343494455739933102218500763, 3.58569590515414638411341916070, 4.46150960180458677460823007145, 4.69467750720940404804277596242, 5.35185830673477984120739825870, 5.62101924871643653198916000849, 6.42265024675426580789242443153, 6.72867725196154039860846369462, 6.94037299505054646117398591462, 7.931134464477924215164870637458, 8.372993657359898213658539994643, 8.430605539601025991731967657355, 8.859645511320313339269777859808, 8.887186370007727515213713566823, 9.785512387266771973485987118759, 9.982464415011675720684641125250
