| L(s) = 1 | + 3·2-s + 3·3-s + 4·4-s + 3·5-s + 9·6-s + 3·8-s + 6·9-s + 9·10-s − 3·11-s + 12·12-s − 3·13-s + 9·15-s + 3·16-s − 6·17-s + 18·18-s + 12·20-s − 9·22-s − 9·23-s + 9·24-s + 5·25-s − 9·26-s + 9·27-s − 9·29-s + 27·30-s + 6·31-s + 6·32-s − 9·33-s + ⋯ |
| L(s) = 1 | + 2.12·2-s + 1.73·3-s + 2·4-s + 1.34·5-s + 3.67·6-s + 1.06·8-s + 2·9-s + 2.84·10-s − 0.904·11-s + 3.46·12-s − 0.832·13-s + 2.32·15-s + 3/4·16-s − 1.45·17-s + 4.24·18-s + 2.68·20-s − 1.91·22-s − 1.87·23-s + 1.83·24-s + 25-s − 1.76·26-s + 1.73·27-s − 1.67·29-s + 4.92·30-s + 1.07·31-s + 1.06·32-s − 1.56·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 194481 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 194481 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(9.957500127\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.957500127\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2$ | \( 1 - p T + p T^{2} \) |
| 7 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 - 3 T + 5 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 3 T + 4 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 3 T + 14 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 9 T + 50 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 9 T + 56 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 6 T + 43 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 3 T - 32 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + T - 42 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 31 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 24 T + 253 T^{2} + 24 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 4 T - 51 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 119 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 8 T - 15 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 15 T + 142 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 3 T + 100 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
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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
−11.34582238783095107570510144196, −11.09266146440327756123774117088, −10.19662038612143042486183200387, −10.03750121559557998738318268967, −9.605731589526578856957729380181, −9.188685504398509622875577789667, −8.602417367992486375539587891537, −8.094850453739430088337381461152, −7.52933434841749556807759368692, −7.29091867727508752023097414247, −6.38318209288207737168864950574, −5.81583506044143123376868560306, −5.81463287395764328083574670003, −4.78191485967828416614906858455, −4.38748088071516192179464386296, −4.26939202468759903346873530778, −3.23022382706163647459272955040, −2.90617542128340820678009772787, −2.12049134362405824669527275438, −1.98618662673246711139340211545,
1.98618662673246711139340211545, 2.12049134362405824669527275438, 2.90617542128340820678009772787, 3.23022382706163647459272955040, 4.26939202468759903346873530778, 4.38748088071516192179464386296, 4.78191485967828416614906858455, 5.81463287395764328083574670003, 5.81583506044143123376868560306, 6.38318209288207737168864950574, 7.29091867727508752023097414247, 7.52933434841749556807759368692, 8.094850453739430088337381461152, 8.602417367992486375539587891537, 9.188685504398509622875577789667, 9.605731589526578856957729380181, 10.03750121559557998738318268967, 10.19662038612143042486183200387, 11.09266146440327756123774117088, 11.34582238783095107570510144196
