| L(s) = 1 | + 2-s + 3-s + 2·4-s − 4·5-s + 6-s − 4·7-s + 5·8-s − 4·10-s + 4·11-s + 2·12-s − 4·14-s − 4·15-s + 5·16-s − 2·17-s − 8·20-s − 4·21-s + 4·22-s + 5·24-s + 2·25-s − 27-s − 8·28-s + 10·29-s − 4·30-s − 8·31-s + 10·32-s + 4·33-s − 2·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 4-s − 1.78·5-s + 0.408·6-s − 1.51·7-s + 1.76·8-s − 1.26·10-s + 1.20·11-s + 0.577·12-s − 1.06·14-s − 1.03·15-s + 5/4·16-s − 0.485·17-s − 1.78·20-s − 0.872·21-s + 0.852·22-s + 1.02·24-s + 2/5·25-s − 0.192·27-s − 1.51·28-s + 1.85·29-s − 0.730·30-s − 1.43·31-s + 1.76·32-s + 0.696·33-s − 0.342·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 257049 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 257049 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.523370347\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.523370347\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 13 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 - T - T^{2} - p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 4 T + 5 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 2 T - 13 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 10 T + 71 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 2 T - 33 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 6 T - 5 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 12 T + 101 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 12 T + 85 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 2 T - 57 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 8 T - 3 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 2 T - 85 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 10 T + 3 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
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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
−11.20364562420934786870735072249, −10.91082762085108215318468509349, −10.25070766310604464622937920554, −9.929326569523744335718636515994, −9.356319336327605337714852632156, −8.675479495880407507416020894278, −8.584221031990018921512585595383, −7.72219644707522635571244564930, −7.41413062405619755433591484092, −7.16848144627570725278851980456, −6.60603110459454829388153230802, −6.19417251910214000081290352133, −5.61031594342196064957574046837, −4.65206823403212734468833470923, −4.23575502597630640115344810438, −3.73668906108316903959501848053, −3.58577020098926599084787960698, −2.73303667353826011515671348488, −2.12317333635998849505750016637, −0.839282436912720068322739163135,
0.839282436912720068322739163135, 2.12317333635998849505750016637, 2.73303667353826011515671348488, 3.58577020098926599084787960698, 3.73668906108316903959501848053, 4.23575502597630640115344810438, 4.65206823403212734468833470923, 5.61031594342196064957574046837, 6.19417251910214000081290352133, 6.60603110459454829388153230802, 7.16848144627570725278851980456, 7.41413062405619755433591484092, 7.72219644707522635571244564930, 8.584221031990018921512585595383, 8.675479495880407507416020894278, 9.356319336327605337714852632156, 9.929326569523744335718636515994, 10.25070766310604464622937920554, 10.91082762085108215318468509349, 11.20364562420934786870735072249
