| L(s) = 1 | + 3·3-s + 5·5-s + 2·9-s − 2·13-s + 15·15-s + 2·17-s + 4·19-s + 6·23-s + 10·25-s − 6·27-s + 10·29-s − 31-s − 6·37-s − 6·39-s − 9·41-s + 3·43-s + 10·45-s + 4·47-s + 6·51-s + 9·53-s + 12·57-s + 6·59-s + 5·61-s − 10·65-s + 67-s + 18·69-s − 14·71-s + ⋯ |
| L(s) = 1 | + 1.73·3-s + 2.23·5-s + 2/3·9-s − 0.554·13-s + 3.87·15-s + 0.485·17-s + 0.917·19-s + 1.25·23-s + 2·25-s − 1.15·27-s + 1.85·29-s − 0.179·31-s − 0.986·37-s − 0.960·39-s − 1.40·41-s + 0.457·43-s + 1.49·45-s + 0.583·47-s + 0.840·51-s + 1.23·53-s + 1.58·57-s + 0.781·59-s + 0.640·61-s − 1.24·65-s + 0.122·67-s + 2.16·69-s − 1.66·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11102224 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11102224 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(9.259006692\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.259006692\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | | \( 1 \) |
| 17 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 - p T + 7 T^{2} - p^{2} T^{3} + p^{2} T^{4} \) |
| 5 | $C_4$ | \( 1 - p T + 3 p T^{2} - p^{2} T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $D_{4}$ | \( 1 + 2 T + 22 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 4 T + 22 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 6 T + 50 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 10 T + 78 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_4$ | \( 1 + T - 39 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 6 T + 38 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 9 T + 91 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 3 T + 77 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 9 T + 65 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 6 T + 122 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 5 T + 27 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - T + 103 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 14 T + 186 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - T + 135 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 10 T + 138 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 12 T + 182 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 22 T + 294 T^{2} - 22 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - T + 183 T^{2} - p T^{3} + p^{2} T^{4} \) |
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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
−8.879013673811184043608678615488, −8.724832633151008969326041321433, −8.016424666891161792444779543259, −7.939841840103145987156685408306, −7.21255838740300075668991186632, −7.15923958658699610569064113328, −6.47961626777573312058944839933, −6.29904525058262122446691698769, −5.63277144071430641358938984762, −5.52164488398356962041781308158, −4.91911535703869940747229027303, −4.88997697047779949498382368269, −3.94229319795948761347162377097, −3.50082886072682046288404034446, −3.07165925805224840936043531837, −2.73815711541078551966990732591, −2.33897299870422614144443771330, −2.02095606936778740314664235010, −1.43154103520227827629246674825, −0.820921076811590809590745864719,
0.820921076811590809590745864719, 1.43154103520227827629246674825, 2.02095606936778740314664235010, 2.33897299870422614144443771330, 2.73815711541078551966990732591, 3.07165925805224840936043531837, 3.50082886072682046288404034446, 3.94229319795948761347162377097, 4.88997697047779949498382368269, 4.91911535703869940747229027303, 5.52164488398356962041781308158, 5.63277144071430641358938984762, 6.29904525058262122446691698769, 6.47961626777573312058944839933, 7.15923958658699610569064113328, 7.21255838740300075668991186632, 7.939841840103145987156685408306, 8.016424666891161792444779543259, 8.724832633151008969326041321433, 8.879013673811184043608678615488
