L(s) = 1 | + (−1.42 − 1.42i)2-s + (1.25 − 0.721i)3-s + 2.06i·4-s + (3.16 + 0.849i)5-s + (−2.81 − 0.753i)6-s + (−0.111 − 2.64i)7-s + (0.0872 − 0.0872i)8-s + (−0.457 + 0.793i)9-s + (−3.30 − 5.72i)10-s + (−5.74 − 1.53i)11-s + (1.48 + 2.57i)12-s + (2.81 + 2.25i)13-s + (−3.60 + 3.92i)14-s + (4.57 − 1.22i)15-s + 3.87·16-s − 0.628·17-s + ⋯ |
L(s) = 1 | + (−1.00 − 1.00i)2-s + (0.721 − 0.416i)3-s + 1.03i·4-s + (1.41 + 0.379i)5-s + (−1.14 − 0.307i)6-s + (−0.0421 − 0.999i)7-s + (0.0308 − 0.0308i)8-s + (−0.152 + 0.264i)9-s + (−1.04 − 1.81i)10-s + (−1.73 − 0.463i)11-s + (0.429 + 0.743i)12-s + (0.779 + 0.626i)13-s + (−0.964 + 1.04i)14-s + (1.18 − 0.316i)15-s + 0.968·16-s − 0.152·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0411 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0411 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.601921 - 0.577626i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.601921 - 0.577626i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 + (0.111 + 2.64i)T \) |
| 13 | \( 1 + (-2.81 - 2.25i)T \) |
good | 2 | \( 1 + (1.42 + 1.42i)T + 2iT^{2} \) |
| 3 | \( 1 + (-1.25 + 0.721i)T + (1.5 - 2.59i)T^{2} \) |
| 5 | \( 1 + (-3.16 - 0.849i)T + (4.33 + 2.5i)T^{2} \) |
| 11 | \( 1 + (5.74 + 1.53i)T + (9.52 + 5.5i)T^{2} \) |
| 17 | \( 1 + 0.628T + 17T^{2} \) |
| 19 | \( 1 + (0.191 + 0.712i)T + (-16.4 + 9.5i)T^{2} \) |
| 23 | \( 1 - 4.54iT - 23T^{2} \) |
| 29 | \( 1 + (1.33 - 2.31i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (0.285 + 1.06i)T + (-26.8 + 15.5i)T^{2} \) |
| 37 | \( 1 + (-1.79 + 1.79i)T - 37iT^{2} \) |
| 41 | \( 1 + (0.746 + 2.78i)T + (-35.5 + 20.5i)T^{2} \) |
| 43 | \( 1 + (-3.49 + 2.01i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (1.90 - 7.10i)T + (-40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (-3.89 + 6.73i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-0.673 - 0.673i)T + 59iT^{2} \) |
| 61 | \( 1 + (0.943 + 0.544i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (1.39 - 5.21i)T + (-58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 + (-0.590 + 2.20i)T + (-61.4 - 35.5i)T^{2} \) |
| 73 | \( 1 + (5.94 - 1.59i)T + (63.2 - 36.5i)T^{2} \) |
| 79 | \( 1 + (6.08 + 10.5i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-3.59 + 3.59i)T - 83iT^{2} \) |
| 89 | \( 1 + (2.88 + 2.88i)T + 89iT^{2} \) |
| 97 | \( 1 + (-8.41 - 2.25i)T + (84.0 + 48.5i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.52599844218735111783735557383, −13.11818395866057672586430332385, −11.10414016409495126351615936127, −10.55914964983302306544133087756, −9.605890451210837088978532108451, −8.528311893859910551479201769517, −7.42129238259283802058873636811, −5.67521263148767595979259355703, −3.00585260621055337134555787911, −1.83410754671187552947916835271,
2.63506599991673285221422834252, 5.39803121277205079689560063600, 6.24716863177539271612242046079, 8.050082379385469964947032260323, 8.763353389606734682430584552827, 9.623427539153283328734730567076, 10.39194482845576537803820426439, 12.55325899098142292771252282247, 13.45301076112332470216686834473, 14.80051200085141983666658530904