| L(s) = 1 | − 5-s − 3·7-s − 2·11-s − 5·13-s + 5·17-s + 7·19-s + 5·23-s − 25-s − 3·29-s − 7·31-s + 3·35-s − 6·37-s + 12·41-s + 8·43-s + 3·47-s + 49-s − 10·53-s + 2·55-s − 14·59-s + 61-s + 5·65-s + 4·67-s − 8·71-s − 7·73-s + 6·77-s + 7·79-s − 25·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 1.13·7-s − 0.603·11-s − 1.38·13-s + 1.21·17-s + 1.60·19-s + 1.04·23-s − 1/5·25-s − 0.557·29-s − 1.25·31-s + 0.507·35-s − 0.986·37-s + 1.87·41-s + 1.21·43-s + 0.437·47-s + 1/7·49-s − 1.37·53-s + 0.269·55-s − 1.82·59-s + 0.128·61-s + 0.620·65-s + 0.488·67-s − 0.949·71-s − 0.819·73-s + 0.683·77-s + 0.787·79-s − 2.74·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 26873856 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26873856 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| good | 5 | $D_{4}$ | \( 1 + T + 2 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 3 T + 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 13 | $D_{4}$ | \( 1 + 5 T + 24 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 5 T + 32 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 7 T + 42 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 5 T + 44 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 3 T + 52 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 7 T + 66 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 6 T + 50 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 12 T + 85 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 8 T + 69 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 3 T + 88 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 10 T + 98 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 61 | $D_{4}$ | \( 1 - T + 114 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 4 T + 105 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 73 | $D_{4}$ | \( 1 + 7 T + 84 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 7 T + 162 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 25 T + 314 T^{2} + 25 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $D_{4}$ | \( 1 + 8 T + 177 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
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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
−7.64007348106205796593241808198, −7.61283064143450920242743909962, −7.47559955258290084660091828201, −7.19456314554595698741024764143, −6.65959372690312992809474308323, −6.14644295942272139182967757016, −5.87368491136012682249948220706, −5.40467406940354682428667715739, −5.19555190343851140708221973366, −4.85970387995768129910052322131, −4.23470855135332796139792595036, −3.90444446486887183161468635022, −3.25091196403307836610796760400, −3.24470748085147118477130749546, −2.60568990188236910599900938087, −2.46918076715863445148485887666, −1.40419665218440545963240562795, −1.16483153304430716729479638739, 0, 0,
1.16483153304430716729479638739, 1.40419665218440545963240562795, 2.46918076715863445148485887666, 2.60568990188236910599900938087, 3.24470748085147118477130749546, 3.25091196403307836610796760400, 3.90444446486887183161468635022, 4.23470855135332796139792595036, 4.85970387995768129910052322131, 5.19555190343851140708221973366, 5.40467406940354682428667715739, 5.87368491136012682249948220706, 6.14644295942272139182967757016, 6.65959372690312992809474308323, 7.19456314554595698741024764143, 7.47559955258290084660091828201, 7.61283064143450920242743909962, 7.64007348106205796593241808198
