| L(s) = 1 | − 2·3-s + 3·4-s − 6·7-s + 2·9-s − 6·11-s − 6·12-s + 5·16-s + 8·17-s + 12·21-s − 2·23-s − 6·27-s − 18·28-s + 6·29-s − 2·31-s + 12·33-s + 6·36-s − 6·37-s − 6·41-s − 18·44-s + 4·47-s − 10·48-s + 18·49-s − 16·51-s + 2·61-s − 12·63-s + 3·64-s + 12·67-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 3/2·4-s − 2.26·7-s + 2/3·9-s − 1.80·11-s − 1.73·12-s + 5/4·16-s + 1.94·17-s + 2.61·21-s − 0.417·23-s − 1.15·27-s − 3.40·28-s + 1.11·29-s − 0.359·31-s + 2.08·33-s + 36-s − 0.986·37-s − 0.937·41-s − 2.71·44-s + 0.583·47-s − 1.44·48-s + 18/7·49-s − 2.24·51-s + 0.256·61-s − 1.51·63-s + 3/8·64-s + 1.46·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 180625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 180625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8263459818\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8263459818\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | | \( 1 \) |
| 17 | $C_2$ | \( 1 - 8 T + p T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - 3 T^{2} + p^{2} T^{4} \) |
| 3 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 58 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 14 T + 98 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 150 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
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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
−11.92351263353758655517547923465, −10.85288534787206863613664780116, −10.52651583056021977505408685280, −10.17638680979767677528936888510, −9.735353948737950290286309512124, −9.637382338936088951922805099041, −8.554667927277463650858942720792, −8.023010886338626353670450989520, −7.52990186095398680341139091394, −7.01810061752613472156754052628, −6.76714988496951488910683409710, −6.11787150208091906409812956620, −5.84959385588803731290290142881, −5.43728157632445237146996716750, −4.90685685004734445533013436417, −3.62961153618714600773447475715, −3.34266943572032839851220758708, −2.74580015880356677509674999510, −2.02401108121806693620180645918, −0.58750809406244438518557466809,
0.58750809406244438518557466809, 2.02401108121806693620180645918, 2.74580015880356677509674999510, 3.34266943572032839851220758708, 3.62961153618714600773447475715, 4.90685685004734445533013436417, 5.43728157632445237146996716750, 5.84959385588803731290290142881, 6.11787150208091906409812956620, 6.76714988496951488910683409710, 7.01810061752613472156754052628, 7.52990186095398680341139091394, 8.023010886338626353670450989520, 8.554667927277463650858942720792, 9.637382338936088951922805099041, 9.735353948737950290286309512124, 10.17638680979767677528936888510, 10.52651583056021977505408685280, 10.85288534787206863613664780116, 11.92351263353758655517547923465
