| L(s) = 1 | + (−0.809 + 0.587i)2-s + (−0.102 + 0.314i)3-s + (0.309 − 0.951i)4-s + (2.08 + 0.799i)5-s + (−0.102 − 0.314i)6-s + 4.37·7-s + (0.309 + 0.951i)8-s + (2.33 + 1.69i)9-s + (−2.15 + 0.580i)10-s + (1.51 − 1.10i)11-s + (0.267 + 0.194i)12-s + (−0.809 − 0.587i)13-s + (−3.54 + 2.57i)14-s + (−0.465 + 0.575i)15-s + (−0.809 − 0.587i)16-s + (−1.34 − 4.14i)17-s + ⋯ |
| L(s) = 1 | + (−0.572 + 0.415i)2-s + (−0.0590 + 0.181i)3-s + (0.154 − 0.475i)4-s + (0.933 + 0.357i)5-s + (−0.0417 − 0.128i)6-s + 1.65·7-s + (0.109 + 0.336i)8-s + (0.779 + 0.566i)9-s + (−0.682 + 0.183i)10-s + (0.457 − 0.332i)11-s + (0.0772 + 0.0561i)12-s + (−0.224 − 0.163i)13-s + (−0.946 + 0.687i)14-s + (−0.120 + 0.148i)15-s + (−0.202 − 0.146i)16-s + (−0.326 − 1.00i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.795 - 0.605i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.795 - 0.605i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.53837 + 0.519170i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.53837 + 0.519170i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.809 - 0.587i)T \) |
| 5 | \( 1 + (-2.08 - 0.799i)T \) |
| 13 | \( 1 + (0.809 + 0.587i)T \) |
| good | 3 | \( 1 + (0.102 - 0.314i)T + (-2.42 - 1.76i)T^{2} \) |
| 7 | \( 1 - 4.37T + 7T^{2} \) |
| 11 | \( 1 + (-1.51 + 1.10i)T + (3.39 - 10.4i)T^{2} \) |
| 17 | \( 1 + (1.34 + 4.14i)T + (-13.7 + 9.99i)T^{2} \) |
| 19 | \( 1 + (1.89 + 5.82i)T + (-15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + (3.00 - 2.18i)T + (7.10 - 21.8i)T^{2} \) |
| 29 | \( 1 + (-0.545 + 1.67i)T + (-23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (0.796 + 2.45i)T + (-25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (-1.42 - 1.03i)T + (11.4 + 35.1i)T^{2} \) |
| 41 | \( 1 + (3.91 + 2.84i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 - 0.0361T + 43T^{2} \) |
| 47 | \( 1 + (2.97 - 9.15i)T + (-38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (3.60 - 11.1i)T + (-42.8 - 31.1i)T^{2} \) |
| 59 | \( 1 + (1.63 + 1.18i)T + (18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (7.90 - 5.74i)T + (18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + (3.19 + 9.83i)T + (-54.2 + 39.3i)T^{2} \) |
| 71 | \( 1 + (-0.130 + 0.400i)T + (-57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (0.159 - 0.115i)T + (22.5 - 69.4i)T^{2} \) |
| 79 | \( 1 + (-5.21 + 16.0i)T + (-63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (-4.27 - 13.1i)T + (-67.1 + 48.7i)T^{2} \) |
| 89 | \( 1 + (0.610 - 0.443i)T + (27.5 - 84.6i)T^{2} \) |
| 97 | \( 1 + (5.22 - 16.0i)T + (-78.4 - 57.0i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.77971408657510649450543464207, −9.618509100162600510908640501594, −9.040871874114205579328355078891, −7.920190069911021311916102263512, −7.27201152385268471254100854850, −6.24441213757744197197611678514, −5.13343947851639222590610801407, −4.52611675658213226568187862941, −2.46829420355440260557378419889, −1.43435407945762717931597519980,
1.52379553938584711591346617463, 1.89770583372004201883629648300, 3.93034881189527200664962714476, 4.82007388782336855695795044507, 6.05527163560207688814186543238, 6.99138916809369729658159715931, 8.164880182268613398880514742883, 8.631142274024630254480934040254, 9.771715856839530243390840084967, 10.30065052890629695641983839593
