| L(s) = 1 | − 4-s + 4·11-s + 16-s − 16·19-s + 18·29-s + 2·31-s − 22·41-s − 4·44-s − 49-s + 14·59-s − 10·61-s − 64-s + 8·71-s + 16·76-s + 12·79-s − 20·89-s − 8·109-s − 18·116-s − 10·121-s − 2·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + ⋯ |
| L(s) = 1 | − 1/2·4-s + 1.20·11-s + 1/4·16-s − 3.67·19-s + 3.34·29-s + 0.359·31-s − 3.43·41-s − 0.603·44-s − 1/7·49-s + 1.82·59-s − 1.28·61-s − 1/8·64-s + 0.949·71-s + 1.83·76-s + 1.35·79-s − 2.11·89-s − 0.766·109-s − 1.67·116-s − 0.909·121-s − 0.179·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9922500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9922500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.158524722\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.158524722\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + T^{2} \) |
| good | 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 23 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 15 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 21 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 11 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 77 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 78 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 97 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 85 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 94 T^{2} + p^{2} T^{4} \) |
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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
−8.853275170807142537891153385401, −8.359063473281505591935829106303, −8.340033520648519684455229869048, −8.101684733450376567932985661289, −7.25166980335507169927531130144, −6.73277316906065924708901819067, −6.61436164253623202977620184795, −6.32957002532125918216990431556, −6.14202258076991458656565764398, −5.19463959774859308182043047669, −5.04970164333507718777609582642, −4.47297915487857364706196279247, −4.32464660042979915556858373212, −3.76511834172431116782694093027, −3.51648091045218432085558216112, −2.62588015808960914988669655533, −2.46935528906122070996103266726, −1.67568919804336700782523855238, −1.26326178822355029715871554354, −0.34920238595241510993526294456,
0.34920238595241510993526294456, 1.26326178822355029715871554354, 1.67568919804336700782523855238, 2.46935528906122070996103266726, 2.62588015808960914988669655533, 3.51648091045218432085558216112, 3.76511834172431116782694093027, 4.32464660042979915556858373212, 4.47297915487857364706196279247, 5.04970164333507718777609582642, 5.19463959774859308182043047669, 6.14202258076991458656565764398, 6.32957002532125918216990431556, 6.61436164253623202977620184795, 6.73277316906065924708901819067, 7.25166980335507169927531130144, 8.101684733450376567932985661289, 8.340033520648519684455229869048, 8.359063473281505591935829106303, 8.853275170807142537891153385401
