| L(s) = 1 | − 3-s − 2·5-s + 3·7-s − 9-s − 2·13-s + 2·15-s − 7·17-s + 9·19-s − 3·21-s − 6·23-s + 3·25-s + 5·29-s − 5·31-s − 6·35-s − 37-s + 2·39-s + 20·41-s + 2·43-s + 2·45-s + 6·47-s − 3·49-s + 7·51-s − 13·53-s − 9·57-s + 2·59-s + 3·61-s − 3·63-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.894·5-s + 1.13·7-s − 1/3·9-s − 0.554·13-s + 0.516·15-s − 1.69·17-s + 2.06·19-s − 0.654·21-s − 1.25·23-s + 3/5·25-s + 0.928·29-s − 0.898·31-s − 1.01·35-s − 0.164·37-s + 0.320·39-s + 3.12·41-s + 0.304·43-s + 0.298·45-s + 0.875·47-s − 3/7·49-s + 0.980·51-s − 1.78·53-s − 1.19·57-s + 0.260·59-s + 0.384·61-s − 0.377·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 93702400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 93702400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.698915351\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.698915351\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 11 | | \( 1 \) |
| good | 3 | $D_{4}$ | \( 1 + T + 2 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 3 T + 12 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_4$ | \( 1 + 2 T + 10 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 7 T + 42 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 9 T + 54 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 6 T + 38 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 5 T + 60 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 5 T + 30 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + T - 32 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 43 | $D_{4}$ | \( 1 - 2 T + 70 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 6 T + 86 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 13 T + 144 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 2 T - 34 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 3 T + 120 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 71 | $D_{4}$ | \( 1 - 9 T + 158 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 18 T + 210 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 10 T + 166 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 89 | $D_{4}$ | \( 1 + T + 72 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 26 T + 346 T^{2} - 26 p T^{3} + p^{2} T^{4} \) |
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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
−7.71757776877214733658824471252, −7.48592897308533257220503072844, −7.41610614001904489919409690438, −6.91514263576475299516286763907, −6.34617082475969521303939301253, −6.21717514072273057621136537537, −5.70220538958208522503390394776, −5.57350002110435635509318167587, −4.83328659809043794188609148431, −4.73666520752694850221683402486, −4.65969024370817705739210529722, −4.07204203530136266060364144131, −3.57380954395202073141426774481, −3.43444997252985248498818343085, −2.69108842147570682530367953226, −2.39422471113254328376395343192, −2.00538506219869449645960073331, −1.41137435669781449557502064742, −0.78608357438396380565795427270, −0.40932763667768038606561334184,
0.40932763667768038606561334184, 0.78608357438396380565795427270, 1.41137435669781449557502064742, 2.00538506219869449645960073331, 2.39422471113254328376395343192, 2.69108842147570682530367953226, 3.43444997252985248498818343085, 3.57380954395202073141426774481, 4.07204203530136266060364144131, 4.65969024370817705739210529722, 4.73666520752694850221683402486, 4.83328659809043794188609148431, 5.57350002110435635509318167587, 5.70220538958208522503390394776, 6.21717514072273057621136537537, 6.34617082475969521303939301253, 6.91514263576475299516286763907, 7.41610614001904489919409690438, 7.48592897308533257220503072844, 7.71757776877214733658824471252
