| L(s) = 1 | − 2·2-s + 2·3-s + 3·4-s − 2·5-s − 4·6-s − 3·7-s − 4·8-s + 3·9-s + 4·10-s + 6·12-s + 6·14-s − 4·15-s + 5·16-s + 2·17-s − 6·18-s + 3·19-s − 6·20-s − 6·21-s − 3·23-s − 8·24-s + 3·25-s + 4·27-s − 9·28-s + 9·29-s + 8·30-s + 31-s − 6·32-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 1.15·3-s + 3/2·4-s − 0.894·5-s − 1.63·6-s − 1.13·7-s − 1.41·8-s + 9-s + 1.26·10-s + 1.73·12-s + 1.60·14-s − 1.03·15-s + 5/4·16-s + 0.485·17-s − 1.41·18-s + 0.688·19-s − 1.34·20-s − 1.30·21-s − 0.625·23-s − 1.63·24-s + 3/5·25-s + 0.769·27-s − 1.70·28-s + 1.67·29-s + 1.46·30-s + 0.179·31-s − 1.06·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25704900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25704900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.804952171\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.804952171\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 3 | $C_1$ | \( ( 1 - T )^{2} \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 13 | | \( 1 \) |
| good | 7 | $D_{4}$ | \( 1 + 3 T + 12 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 5 T^{2} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 2 T + 18 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 3 T + 36 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 3 T + 10 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 9 T + 74 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - T + 24 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 57 T^{2} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 8 T + 30 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 5 T + 88 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 53 | $D_{4}$ | \( 1 - 13 T + 144 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 17 T + 186 T^{2} + 17 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 67 | $D_{4}$ | \( 1 + 12 T + 102 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 18 T + 206 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 6 T + 2 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 19 T + 244 T^{2} - 19 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 6 T + 158 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 17 T + 144 T^{2} - 17 p T^{3} + p^{2} T^{4} \) |
| 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
−8.200759942926475887883342377837, −8.195280959558048294787727422692, −7.81081928090076381189430341138, −7.64590057049480770945303173812, −6.92968501350855486956021431596, −6.89416453824532615166567186733, −6.39287299897593595098012084753, −6.31972551719153185005540522993, −5.54926904692810806299812084300, −5.14535665286131711691840023259, −4.63900621305756525149461604523, −4.20651111371606470529357788444, −3.50164576360656421014559928166, −3.46596048994327775681382637494, −3.00676041040595989783494202626, −2.75283172085853759787916183246, −1.94558476555652728035009906379, −1.77376381524662500176950815549, −0.70152726213898222720360668206, −0.65678732101566693168013401251,
0.65678732101566693168013401251, 0.70152726213898222720360668206, 1.77376381524662500176950815549, 1.94558476555652728035009906379, 2.75283172085853759787916183246, 3.00676041040595989783494202626, 3.46596048994327775681382637494, 3.50164576360656421014559928166, 4.20651111371606470529357788444, 4.63900621305756525149461604523, 5.14535665286131711691840023259, 5.54926904692810806299812084300, 6.31972551719153185005540522993, 6.39287299897593595098012084753, 6.89416453824532615166567186733, 6.92968501350855486956021431596, 7.64590057049480770945303173812, 7.81081928090076381189430341138, 8.195280959558048294787727422692, 8.200759942926475887883342377837
