| L(s) = 1 | − 4-s − 9-s − 6·11-s + 16-s + 2·19-s + 10·29-s + 14·31-s + 36-s + 4·41-s + 6·44-s + 10·49-s + 20·59-s − 26·61-s − 64-s + 24·71-s − 2·76-s − 10·79-s + 81-s − 10·89-s + 6·99-s + 4·101-s − 20·109-s − 10·116-s + 5·121-s − 14·124-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 1/3·9-s − 1.80·11-s + 1/4·16-s + 0.458·19-s + 1.85·29-s + 2.51·31-s + 1/6·36-s + 0.624·41-s + 0.904·44-s + 10/7·49-s + 2.60·59-s − 3.32·61-s − 1/8·64-s + 2.84·71-s − 0.229·76-s − 1.12·79-s + 1/9·81-s − 1.05·89-s + 0.603·99-s + 0.398·101-s − 1.91·109-s − 0.928·116-s + 5/11·121-s − 1.25·124-s + 0.0887·127-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8122500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8122500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.958439271\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.958439271\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 45 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 13 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 85 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 65 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 165 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
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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
−8.876953180679145020259005279157, −8.462279199523224918592703960594, −8.227442610722895579362172838116, −7.921242446631984372756002799708, −7.69332819003604671499366066574, −6.98396662738868629969933946053, −6.79702286144541205171218715456, −6.28027786695121415592994528889, −5.79098249384282533703610905507, −5.53247227159167075779657588176, −4.97362263765330483404157859474, −4.85462015337871508270560184677, −4.21588923682454297543402975749, −4.01830198106976228824132513125, −3.05589714247615099722976532279, −2.87140058729412711720948942795, −2.61166241734859765591218639953, −1.89372479066211228136237285980, −0.931711828038443377798866922169, −0.57507951696368228021810819184,
0.57507951696368228021810819184, 0.931711828038443377798866922169, 1.89372479066211228136237285980, 2.61166241734859765591218639953, 2.87140058729412711720948942795, 3.05589714247615099722976532279, 4.01830198106976228824132513125, 4.21588923682454297543402975749, 4.85462015337871508270560184677, 4.97362263765330483404157859474, 5.53247227159167075779657588176, 5.79098249384282533703610905507, 6.28027786695121415592994528889, 6.79702286144541205171218715456, 6.98396662738868629969933946053, 7.69332819003604671499366066574, 7.921242446631984372756002799708, 8.227442610722895579362172838116, 8.462279199523224918592703960594, 8.876953180679145020259005279157
