| L(s) = 1 | − 4-s − 9-s − 10·11-s + 16-s − 2·19-s − 20·29-s + 10·31-s + 36-s + 12·41-s + 10·44-s + 5·49-s − 16·59-s − 4·61-s − 64-s + 12·71-s + 2·76-s − 10·79-s + 81-s − 24·89-s + 10·99-s + 22·101-s − 10·109-s + 20·116-s + 53·121-s − 10·124-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 1/3·9-s − 3.01·11-s + 1/4·16-s − 0.458·19-s − 3.71·29-s + 1.79·31-s + 1/6·36-s + 1.87·41-s + 1.50·44-s + 5/7·49-s − 2.08·59-s − 0.512·61-s − 1/8·64-s + 1.42·71-s + 0.229·76-s − 1.12·79-s + 1/9·81-s − 2.54·89-s + 1.00·99-s + 2.18·101-s − 0.957·109-s + 1.85·116-s + 4.81·121-s − 0.898·124-s + 0.0887·127-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6502500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6502500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 3 | $C_2$ | \( 1 + T^{2} \) |
| 5 | | \( 1 \) |
| 17 | $C_2$ | \( 1 + T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 65 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 85 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 85 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 105 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 13 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 158 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
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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
−8.753085849492145649410358642210, −8.062297722908699188937248350481, −7.977376926333309118175734280291, −7.74454738505994152014273527758, −7.34613799676728884530216814158, −6.94274765063363805424984972340, −6.20519285789021037069340056440, −5.75379687540167972938753962772, −5.71549906655029950639976440705, −5.13243726669882680950299969767, −4.92920505331257891477187201512, −4.21844450252741656683311538202, −4.07853809484330157121327256064, −3.20650849535453936657773173674, −2.98209019074261041681377611880, −2.31404446576434806501420573190, −2.15227897478228432824090523678, −1.14469489677111027080542182623, 0, 0,
1.14469489677111027080542182623, 2.15227897478228432824090523678, 2.31404446576434806501420573190, 2.98209019074261041681377611880, 3.20650849535453936657773173674, 4.07853809484330157121327256064, 4.21844450252741656683311538202, 4.92920505331257891477187201512, 5.13243726669882680950299969767, 5.71549906655029950639976440705, 5.75379687540167972938753962772, 6.20519285789021037069340056440, 6.94274765063363805424984972340, 7.34613799676728884530216814158, 7.74454738505994152014273527758, 7.977376926333309118175734280291, 8.062297722908699188937248350481, 8.753085849492145649410358642210
