| L(s) = 1 | + 94·7-s − 146·13-s − 92·19-s − 625·25-s − 194·31-s − 4.12e3·37-s + 3.21e3·43-s + 2.40e3·49-s + 1.96e3·61-s − 5.90e3·67-s − 1.70e4·73-s − 7.68e3·79-s − 1.37e4·91-s + 1.88e4·97-s − 1.64e4·103-s + 4.40e4·109-s − 1.46e4·121-s + 127-s + 131-s − 8.64e3·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | + 1.91·7-s − 0.863·13-s − 0.254·19-s − 25-s − 0.201·31-s − 3.01·37-s + 1.73·43-s + 49-s + 0.528·61-s − 1.31·67-s − 3.20·73-s − 1.23·79-s − 1.65·91-s + 1.99·97-s − 1.54·103-s + 3.70·109-s − 121-s + 6.20e−5·127-s + 5.82e−5·131-s − 0.488·133-s + 5.32e−5·137-s + 5.17e−5·139-s + 4.50e−5·149-s + 4.38e−5·151-s + 4.05e−5·157-s + 3.76e−5·163-s + 3.58e−5·167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 104976 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 104976 ^{s/2} \, \Gamma_{\C}(s+2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(2.187599956\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.187599956\) |
| \(L(3)\) |
|
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 | $C_2$ | \( ( 1 - p^{2} T + p^{4} T^{2} )( 1 + p^{2} T + p^{4} T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 71 T + p^{4} T^{2} )( 1 - 23 T + p^{4} T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - p^{2} T + p^{4} T^{2} )( 1 + p^{2} T + p^{4} T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 191 T + p^{4} T^{2} )( 1 + 337 T + p^{4} T^{2} ) \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 46 T + p^{4} T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - p^{2} T + p^{4} T^{2} )( 1 + p^{2} T + p^{4} T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - p^{2} T + p^{4} T^{2} )( 1 + p^{2} T + p^{4} T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 1559 T + p^{4} T^{2} )( 1 + 1753 T + p^{4} T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 2062 T + p^{4} T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - p^{2} T + p^{4} T^{2} )( 1 + p^{2} T + p^{4} T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 3191 T + p^{4} T^{2} )( 1 - 23 T + p^{4} T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - p^{2} T + p^{4} T^{2} )( 1 + p^{2} T + p^{4} T^{2} ) \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 59 | $C_2$ | \( ( 1 - p^{2} T + p^{4} T^{2} )( 1 + p^{2} T + p^{4} T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 7199 T + p^{4} T^{2} )( 1 + 5233 T + p^{4} T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 2903 T + p^{4} T^{2} )( 1 + 8809 T + p^{4} T^{2} ) \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 8542 T + p^{4} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 4679 T + p^{4} T^{2} )( 1 + 12361 T + p^{4} T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - p^{2} T + p^{4} T^{2} )( 1 + p^{2} T + p^{4} T^{2} ) \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 9743 T + p^{4} T^{2} )( 1 - 9071 T + p^{4} T^{2} ) \) |
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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
−11.62211673402984883246434008318, −10.63156277366029214139866775196, −10.41619890118529838989236509930, −9.964052317132145616919301876945, −9.229467641415089542265967079520, −8.625994855220122846867806630663, −8.612951604170013644360341376752, −7.63824079867775720142912564671, −7.60803321889693999437652804560, −7.08271643432965905212898455274, −6.31710300278987614922113144894, −5.53547430072669212961725249524, −5.36337056360720982091495009544, −4.50711550412426078214905642859, −4.40612953814072141996745569462, −3.48154186504963024667381117671, −2.68431778446636784165323661547, −1.71790544221951973007950732851, −1.70592548071076871545317175983, −0.43099311119239613797876049890,
0.43099311119239613797876049890, 1.70592548071076871545317175983, 1.71790544221951973007950732851, 2.68431778446636784165323661547, 3.48154186504963024667381117671, 4.40612953814072141996745569462, 4.50711550412426078214905642859, 5.36337056360720982091495009544, 5.53547430072669212961725249524, 6.31710300278987614922113144894, 7.08271643432965905212898455274, 7.60803321889693999437652804560, 7.63824079867775720142912564671, 8.612951604170013644360341376752, 8.625994855220122846867806630663, 9.229467641415089542265967079520, 9.964052317132145616919301876945, 10.41619890118529838989236509930, 10.63156277366029214139866775196, 11.62211673402984883246434008318
