| L(s) = 1 | + 2·3-s + 4·7-s + 2·13-s − 2·19-s + 8·21-s − 2·27-s − 4·31-s − 10·37-s + 4·39-s − 12·41-s + 4·43-s + 12·47-s − 2·49-s + 18·53-s − 4·57-s + 4·61-s + 10·67-s + 12·71-s + 8·73-s + 8·79-s − 81-s − 24·89-s + 8·91-s − 8·93-s + 2·97-s − 12·101-s + 10·103-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 1.51·7-s + 0.554·13-s − 0.458·19-s + 1.74·21-s − 0.384·27-s − 0.718·31-s − 1.64·37-s + 0.640·39-s − 1.87·41-s + 0.609·43-s + 1.75·47-s − 2/7·49-s + 2.47·53-s − 0.529·57-s + 0.512·61-s + 1.22·67-s + 1.42·71-s + 0.936·73-s + 0.900·79-s − 1/9·81-s − 2.54·89-s + 0.838·91-s − 0.829·93-s + 0.203·97-s − 1.19·101-s + 0.985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 57760000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 57760000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.259006763\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.259006763\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 - 2 T + 4 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 2 T + 24 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 22 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 4 T + 54 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 10 T + 96 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $D_{4}$ | \( 1 - 4 T + 42 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 12 T + 82 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 18 T + 184 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 70 T^{2} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 10 T + 156 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 12 T + 166 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 8 T + 150 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 8 T + 126 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 154 T^{2} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 24 T + 310 T^{2} + 24 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 2 T - 48 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
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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
−8.184117279333769463752723412714, −7.889325726438903159238616725052, −7.31215968811451396352487736798, −7.20289814273498266109499431047, −6.71348595204637031534107314995, −6.49653568166326509078440693125, −5.80738118101595791861275433255, −5.54395953320593624690323702646, −5.23074115733347879785666970741, −4.91952098690499083583962902888, −4.50742078205795462240653246559, −3.89142157533166258170583568292, −3.70798630141481108906072967617, −3.51702835688572630748210485359, −2.68785692148324296686400873187, −2.58180902689360741566359779956, −1.88705711337291267248016446614, −1.83826380802147783647719534327, −1.15132721047124141831873828615, −0.50613423632576486396932258033,
0.50613423632576486396932258033, 1.15132721047124141831873828615, 1.83826380802147783647719534327, 1.88705711337291267248016446614, 2.58180902689360741566359779956, 2.68785692148324296686400873187, 3.51702835688572630748210485359, 3.70798630141481108906072967617, 3.89142157533166258170583568292, 4.50742078205795462240653246559, 4.91952098690499083583962902888, 5.23074115733347879785666970741, 5.54395953320593624690323702646, 5.80738118101595791861275433255, 6.49653568166326509078440693125, 6.71348595204637031534107314995, 7.20289814273498266109499431047, 7.31215968811451396352487736798, 7.889325726438903159238616725052, 8.184117279333769463752723412714
