| L(s) = 1 | − 3-s + 2·5-s − 7-s − 9-s + 2·11-s + 4·13-s − 2·15-s + 3·17-s + 19-s + 21-s + 2·23-s + 3·25-s + 7·29-s + 9·31-s − 2·33-s − 2·35-s + 37-s − 4·39-s + 4·41-s + 4·43-s − 2·45-s − 18·47-s − 9·49-s − 3·51-s + 5·53-s + 4·55-s − 57-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.894·5-s − 0.377·7-s − 1/3·9-s + 0.603·11-s + 1.10·13-s − 0.516·15-s + 0.727·17-s + 0.229·19-s + 0.218·21-s + 0.417·23-s + 3/5·25-s + 1.29·29-s + 1.61·31-s − 0.348·33-s − 0.338·35-s + 0.164·37-s − 0.640·39-s + 0.624·41-s + 0.609·43-s − 0.298·45-s − 2.62·47-s − 9/7·49-s − 0.420·51-s + 0.686·53-s + 0.539·55-s − 0.132·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 193600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 193600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.753532385\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.753532385\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 11 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 + T + 2 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + T + 10 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $D_{4}$ | \( 1 - 3 T + 32 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - T + 34 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 2 T + 30 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 7 T + 32 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 9 T + 78 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - T + 36 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 43 | $D_{4}$ | \( 1 - 4 T + 22 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 18 T + 158 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 5 T + 108 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 6 T + 110 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 17 T + 156 T^{2} - 17 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 4 T + 70 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 5 T + 110 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 79 | $D_{4}$ | \( 1 + 6 T + 14 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 2 T + 150 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 21 T + 284 T^{2} + 21 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 6 T + 186 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
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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
−11.14653637118613550077006571412, −11.14126960236329538149896188774, −10.31578290271914889975618761475, −10.06052030113229562892507737326, −9.610102380261814035787365797523, −9.257017920035817868336162369486, −8.506330913515456389452192318983, −8.353327407504750899931651581573, −7.76906379231101684175086952189, −6.90714749289251182514887539182, −6.42240231639132883898098202212, −6.40508595746747246223743250974, −5.55670259443597983322528411319, −5.44013963577261013972736471656, −4.56056513454979795834442771016, −4.09564095664291042840370305454, −3.04747812934299332212872036183, −2.94530830241539239522897907768, −1.66600432967191578976060225189, −0.963725731828977312567788341922,
0.963725731828977312567788341922, 1.66600432967191578976060225189, 2.94530830241539239522897907768, 3.04747812934299332212872036183, 4.09564095664291042840370305454, 4.56056513454979795834442771016, 5.44013963577261013972736471656, 5.55670259443597983322528411319, 6.40508595746747246223743250974, 6.42240231639132883898098202212, 6.90714749289251182514887539182, 7.76906379231101684175086952189, 8.353327407504750899931651581573, 8.506330913515456389452192318983, 9.257017920035817868336162369486, 9.610102380261814035787365797523, 10.06052030113229562892507737326, 10.31578290271914889975618761475, 11.14126960236329538149896188774, 11.14653637118613550077006571412
