| L(s) = 1 | − 4-s − 9-s − 2·11-s + 16-s − 2·19-s + 18·29-s + 6·31-s + 36-s + 12·41-s + 2·44-s + 10·49-s + 20·59-s − 4·61-s − 64-s + 22·71-s + 2·76-s − 8·79-s + 81-s + 22·89-s + 2·99-s − 18·101-s − 22·109-s − 18·116-s + 3·121-s − 6·124-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 1/3·9-s − 0.603·11-s + 1/4·16-s − 0.458·19-s + 3.34·29-s + 1.07·31-s + 1/6·36-s + 1.87·41-s + 0.301·44-s + 10/7·49-s + 2.60·59-s − 0.512·61-s − 1/8·64-s + 2.61·71-s + 0.229·76-s − 0.900·79-s + 1/9·81-s + 2.33·89-s + 0.201·99-s − 1.79·101-s − 2.10·109-s − 1.67·116-s + 3/11·121-s − 0.538·124-s + 0.0887·127-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2722500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2722500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.070334749\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.070334749\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 45 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 98 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 142 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 117 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 113 T^{2} + p^{2} T^{4} \) |
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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
−9.380916337384363682640155749714, −9.331926077573668040144541648347, −8.632446117571959725090359015289, −8.334008708707609341079254160459, −8.164065062420385757824964606121, −7.76947117916552215640511032737, −7.06881687949970541029421894928, −6.81882399399607080815422885463, −6.25576027940393129713728600658, −6.05808283109154790169988160371, −5.26245114082285247808417889399, −5.19732267317909699367515723016, −4.53618565103136084289019370128, −4.21187573233671195085054989270, −3.75637559150580043737040892299, −2.99148031311685586831307527648, −2.56698741612566383762376626526, −2.28924145533601756702510619332, −1.05547703877661652518865627290, −0.68547688995865285960949750094,
0.68547688995865285960949750094, 1.05547703877661652518865627290, 2.28924145533601756702510619332, 2.56698741612566383762376626526, 2.99148031311685586831307527648, 3.75637559150580043737040892299, 4.21187573233671195085054989270, 4.53618565103136084289019370128, 5.19732267317909699367515723016, 5.26245114082285247808417889399, 6.05808283109154790169988160371, 6.25576027940393129713728600658, 6.81882399399607080815422885463, 7.06881687949970541029421894928, 7.76947117916552215640511032737, 8.164065062420385757824964606121, 8.334008708707609341079254160459, 8.632446117571959725090359015289, 9.331926077573668040144541648347, 9.380916337384363682640155749714
