| L(s) = 1 | − 4·2-s − 6·3-s + 12·4-s − 10·5-s + 24·6-s + 37·7-s − 32·8-s + 27·9-s + 40·10-s − 45·11-s − 72·12-s + 103·13-s − 148·14-s + 60·15-s + 80·16-s − 123·17-s − 108·18-s − 71·19-s − 120·20-s − 222·21-s + 180·22-s − 51·23-s + 192·24-s + 75·25-s − 412·26-s − 108·27-s + 444·28-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 1.15·3-s + 3/2·4-s − 0.894·5-s + 1.63·6-s + 1.99·7-s − 1.41·8-s + 9-s + 1.26·10-s − 1.23·11-s − 1.73·12-s + 2.19·13-s − 2.82·14-s + 1.03·15-s + 5/4·16-s − 1.75·17-s − 1.41·18-s − 0.857·19-s − 1.34·20-s − 2.30·21-s + 1.74·22-s − 0.462·23-s + 1.63·24-s + 3/5·25-s − 3.10·26-s − 0.769·27-s + 2.99·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232100 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_1$ | \( ( 1 + p T )^{2} \) |
| 3 | $C_1$ | \( ( 1 + p T )^{2} \) |
| 5 | $C_1$ | \( ( 1 + p T )^{2} \) |
| 37 | $C_1$ | \( ( 1 - p T )^{2} \) |
| good | 7 | $D_{4}$ | \( 1 - 37 T + 822 T^{2} - 37 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 45 T + 2962 T^{2} + 45 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 103 T + 6840 T^{2} - 103 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 123 T + 8452 T^{2} + 123 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 71 T + 13122 T^{2} + 71 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 51 T + 24778 T^{2} + 51 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 180 T + 43678 T^{2} + 180 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 166 T + 65646 T^{2} - 166 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 318 T + 142498 T^{2} + 318 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 56 T + 106998 T^{2} + 56 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 120 T + 92446 T^{2} - 120 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 213 T - 72260 T^{2} - 213 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 978 T + 550054 T^{2} - 978 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 436 T + 382686 T^{2} - 436 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 326 T + 488670 T^{2} + 326 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 186 T - 173954 T^{2} - 186 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 193 T + 117240 T^{2} - 193 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 728 T + 999774 T^{2} + 728 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 675 T + 1211074 T^{2} + 675 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 177 T + 1407664 T^{2} + 177 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 574 T + 1669290 T^{2} - 574 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−8.832374542265119153740190980827, −8.789828469130859811717569358027, −8.478222559520586405737674287315, −8.130328160529831500057359638640, −7.70712971741164458132306366805, −7.39528305105057194392657332681, −6.67185067142681918422956971665, −6.59099368638367367795788764787, −5.88243660947870796755924301160, −5.56787468364211777333206840943, −4.86630374879396842931680018558, −4.67211340322428143218119774605, −3.81056267516107841241805289750, −3.78225191551332140437086848987, −2.38411447955449702459357857653, −2.23150422972486598757395262365, −1.28497824706916691298222827300, −1.14892633967649782488239806693, 0, 0,
1.14892633967649782488239806693, 1.28497824706916691298222827300, 2.23150422972486598757395262365, 2.38411447955449702459357857653, 3.78225191551332140437086848987, 3.81056267516107841241805289750, 4.67211340322428143218119774605, 4.86630374879396842931680018558, 5.56787468364211777333206840943, 5.88243660947870796755924301160, 6.59099368638367367795788764787, 6.67185067142681918422956971665, 7.39528305105057194392657332681, 7.70712971741164458132306366805, 8.130328160529831500057359638640, 8.478222559520586405737674287315, 8.789828469130859811717569358027, 8.832374542265119153740190980827
