| L(s) = 1 | + 2·2-s + 2·3-s + 3·4-s + 6·5-s + 4·6-s + 4·8-s + 3·9-s + 12·10-s − 2·11-s + 6·12-s + 12·15-s + 5·16-s + 6·18-s − 2·19-s + 18·20-s − 4·22-s + 4·23-s + 8·24-s + 19·25-s + 4·27-s + 10·29-s + 24·30-s + 10·31-s + 6·32-s − 4·33-s + 9·36-s − 8·37-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 1.15·3-s + 3/2·4-s + 2.68·5-s + 1.63·6-s + 1.41·8-s + 9-s + 3.79·10-s − 0.603·11-s + 1.73·12-s + 3.09·15-s + 5/4·16-s + 1.41·18-s − 0.458·19-s + 4.02·20-s − 0.852·22-s + 0.834·23-s + 1.63·24-s + 19/5·25-s + 0.769·27-s + 1.85·29-s + 4.38·30-s + 1.79·31-s + 1.06·32-s − 0.696·33-s + 3/2·36-s − 1.31·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 31203396 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31203396 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(28.71504004\) |
| \(L(\frac12)\) |
\(\approx\) |
\(28.71504004\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 3 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 6 T + 17 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 16 T^{2} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 4 T + 48 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 10 T + 75 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 10 T + 55 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 8 T + 72 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 4 T + 84 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 4 T + 72 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 12 T + 122 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 2 T + 99 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 2 T + 69 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 4 T + 118 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 71 | $D_{4}$ | \( 1 - 4 T + 48 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 24 T + 282 T^{2} - 24 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 6 T + 159 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 6 T - 25 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 16 T + 224 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 14 T + 241 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
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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
−8.427138323035753333615975731448, −7.957083199904697693981477737594, −7.49314021466716576775825522846, −7.12468328331029410010459424238, −6.58448763772222345236126053601, −6.51414938393302704117793400698, −6.08043676093827681145388475942, −5.98714753020526358371714885899, −5.18381745550539759255671621656, −5.13367378482480655597351647413, −4.68075793485890613327447783090, −4.59640216056940052385702582253, −3.59470668430850908325768896033, −3.55737725229738493771909416514, −2.77722797000728256453292573127, −2.69873100326959008744893713363, −2.20662443348891781610340106592, −2.09191329426343172937934388991, −1.29342874336370697819299494079, −1.06197005980815594590821971451,
1.06197005980815594590821971451, 1.29342874336370697819299494079, 2.09191329426343172937934388991, 2.20662443348891781610340106592, 2.69873100326959008744893713363, 2.77722797000728256453292573127, 3.55737725229738493771909416514, 3.59470668430850908325768896033, 4.59640216056940052385702582253, 4.68075793485890613327447783090, 5.13367378482480655597351647413, 5.18381745550539759255671621656, 5.98714753020526358371714885899, 6.08043676093827681145388475942, 6.51414938393302704117793400698, 6.58448763772222345236126053601, 7.12468328331029410010459424238, 7.49314021466716576775825522846, 7.957083199904697693981477737594, 8.427138323035753333615975731448
