| L(s) = 1 | − 2-s − 3-s + 5-s + 6-s + 5·7-s + 8-s + 3·9-s − 10-s − 11-s − 2·13-s − 5·14-s − 15-s − 16-s − 3·18-s − 2·19-s − 5·21-s + 22-s + 6·23-s − 24-s + 2·26-s − 8·27-s − 18·29-s + 30-s + 4·31-s + 33-s + 5·35-s + 4·37-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.447·5-s + 0.408·6-s + 1.88·7-s + 0.353·8-s + 9-s − 0.316·10-s − 0.301·11-s − 0.554·13-s − 1.33·14-s − 0.258·15-s − 1/4·16-s − 0.707·18-s − 0.458·19-s − 1.09·21-s + 0.213·22-s + 1.25·23-s − 0.204·24-s + 0.392·26-s − 1.53·27-s − 3.34·29-s + 0.182·30-s + 0.718·31-s + 0.174·33-s + 0.845·35-s + 0.657·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 592900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 592900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.387605489\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.387605489\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T + T^{2} \) |
| 5 | $C_2$ | \( 1 - T + T^{2} \) |
| 7 | $C_2$ | \( 1 - 5 T + p T^{2} \) |
| 11 | $C_2$ | \( 1 + T + T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + T - 2 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 2 T - 15 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 6 T + 13 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 - 4 T - 21 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 12 T + 97 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 6 T - 17 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 3 T - 50 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 7 T - 12 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 13 T + 102 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 14 T + 123 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 6 T - 53 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
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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
−10.61449610944512178646940220150, −10.07796453091352308826375609508, −9.679683399032555341836053891848, −9.139125322022559606312743260315, −9.090802354248279677439902182566, −8.367684695295255592829198576795, −7.83274350919790756672590506936, −7.51789791286739638763228651404, −7.37561853487199537228550294581, −6.79140284589459395358371390830, −5.84137456448527130638979387294, −5.81734617411844556639530652676, −4.97021534511446601711356865089, −4.94751381703905328827684690702, −4.12497470268647556198418499769, −3.91109391330071955111102378890, −2.68931844305498549818898702149, −1.96923828456132310031056932113, −1.64494880361374577318458564940, −0.73395867792172558887207203549,
0.73395867792172558887207203549, 1.64494880361374577318458564940, 1.96923828456132310031056932113, 2.68931844305498549818898702149, 3.91109391330071955111102378890, 4.12497470268647556198418499769, 4.94751381703905328827684690702, 4.97021534511446601711356865089, 5.81734617411844556639530652676, 5.84137456448527130638979387294, 6.79140284589459395358371390830, 7.37561853487199537228550294581, 7.51789791286739638763228651404, 7.83274350919790756672590506936, 8.367684695295255592829198576795, 9.090802354248279677439902182566, 9.139125322022559606312743260315, 9.679683399032555341836053891848, 10.07796453091352308826375609508, 10.61449610944512178646940220150
