| L(s) = 1 | + 3-s + 9-s + 2·13-s − 4·16-s + 2·19-s − 6·25-s + 27-s + 18·31-s + 6·37-s + 2·39-s + 10·43-s − 4·48-s + 2·57-s + 20·61-s − 10·67-s − 6·73-s − 6·75-s − 2·79-s + 81-s + 18·93-s − 12·97-s − 14·103-s + 18·109-s + 6·111-s + 2·117-s − 18·121-s + 127-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1/3·9-s + 0.554·13-s − 16-s + 0.458·19-s − 6/5·25-s + 0.192·27-s + 3.23·31-s + 0.986·37-s + 0.320·39-s + 1.52·43-s − 0.577·48-s + 0.264·57-s + 2.56·61-s − 1.22·67-s − 0.702·73-s − 0.692·75-s − 0.225·79-s + 1/9·81-s + 1.86·93-s − 1.21·97-s − 1.37·103-s + 1.72·109-s + 0.569·111-s + 0.184·117-s − 1.63·121-s + 0.0887·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 64827 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64827 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.788277919\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.788277919\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 - T \) |
| 7 | | \( 1 \) |
| good | 2 | $C_2$ | \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
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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
−9.879400023159387748546449740681, −9.459031124333216147077448358277, −8.693773060427938478692796817657, −8.583706845901838358159961836152, −7.67583845212820398533569804897, −7.67312246287158311998952783143, −6.65130832285966559117996014878, −6.39199342358304616891806651452, −5.72037257717675019727399870073, −4.96455995261269316221263952539, −4.23783192793666623809357478808, −3.92756051958450667801160124862, −2.81502084362613660119824052187, −2.43327032489084752763662196914, −1.15383453307155328641933378041,
1.15383453307155328641933378041, 2.43327032489084752763662196914, 2.81502084362613660119824052187, 3.92756051958450667801160124862, 4.23783192793666623809357478808, 4.96455995261269316221263952539, 5.72037257717675019727399870073, 6.39199342358304616891806651452, 6.65130832285966559117996014878, 7.67312246287158311998952783143, 7.67583845212820398533569804897, 8.583706845901838358159961836152, 8.693773060427938478692796817657, 9.459031124333216147077448358277, 9.879400023159387748546449740681
