L(s) = 1 | + (−0.365 + 0.930i)2-s + (−0.149 + 1.98i)3-s + (−0.733 − 0.680i)4-s + (−1.79 − 0.865i)6-s + (−0.433 − 0.900i)7-s + (0.900 − 0.433i)8-s + (−2.94 − 0.443i)9-s + (1.84 − 0.277i)11-s + (1.46 − 1.35i)12-s + (−0.455 − 0.571i)13-s + (0.997 − 0.0747i)14-s + (0.0747 + 0.997i)16-s + (0.955 + 0.294i)17-s + (1.48 − 2.57i)18-s + (1.85 − 0.728i)21-s + (−0.414 + 1.81i)22-s + ⋯ |
L(s) = 1 | + (−0.365 + 0.930i)2-s + (−0.149 + 1.98i)3-s + (−0.733 − 0.680i)4-s + (−1.79 − 0.865i)6-s + (−0.433 − 0.900i)7-s + (0.900 − 0.433i)8-s + (−2.94 − 0.443i)9-s + (1.84 − 0.277i)11-s + (1.46 − 1.35i)12-s + (−0.455 − 0.571i)13-s + (0.997 − 0.0747i)14-s + (0.0747 + 0.997i)16-s + (0.955 + 0.294i)17-s + (1.48 − 2.57i)18-s + (1.85 − 0.728i)21-s + (−0.414 + 1.81i)22-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.481 - 0.876i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.481 - 0.876i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9249834998\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9249834998\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.365 - 0.930i)T \) |
| 7 | \( 1 + (0.433 + 0.900i)T \) |
| 17 | \( 1 + (-0.955 - 0.294i)T \) |
good | 3 | \( 1 + (0.149 - 1.98i)T + (-0.988 - 0.149i)T^{2} \) |
| 5 | \( 1 + (-0.365 - 0.930i)T^{2} \) |
| 11 | \( 1 + (-1.84 + 0.277i)T + (0.955 - 0.294i)T^{2} \) |
| 13 | \( 1 + (0.455 + 0.571i)T + (-0.222 + 0.974i)T^{2} \) |
| 19 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + (-1.49 + 0.460i)T + (0.826 - 0.563i)T^{2} \) |
| 29 | \( 1 + (0.900 - 0.433i)T^{2} \) |
| 31 | \( 1 + (-0.433 + 0.751i)T + (-0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + (-0.0747 + 0.997i)T^{2} \) |
| 41 | \( 1 + (-0.623 + 0.781i)T^{2} \) |
| 43 | \( 1 + (-0.623 - 0.781i)T^{2} \) |
| 47 | \( 1 + (0.733 + 0.680i)T^{2} \) |
| 53 | \( 1 + (1.21 + 1.12i)T + (0.0747 + 0.997i)T^{2} \) |
| 59 | \( 1 + (-0.365 + 0.930i)T^{2} \) |
| 61 | \( 1 + (-0.0747 + 0.997i)T^{2} \) |
| 67 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 + (-0.250 + 1.09i)T + (-0.900 - 0.433i)T^{2} \) |
| 73 | \( 1 + (0.733 - 0.680i)T^{2} \) |
| 79 | \( 1 + (-0.149 - 0.258i)T + (-0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + (0.222 + 0.974i)T^{2} \) |
| 89 | \( 1 + (-0.147 - 0.0222i)T + (0.955 + 0.294i)T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−9.191474060021538723185689281640, −8.556485924309393959996671871375, −7.63924988433357425389075627848, −6.63020090545962215425442197434, −6.05739409530682606607523932561, −5.14783706370095813021331311642, −4.57284103610135599657559904490, −3.72050301161578072520764466380, −3.27379726588034037706103030928, −0.884543582747539532695854431845,
1.03774895530056787745488036532, 1.71231673097765192992542225312, 2.69026325392106622771774940414, 3.31606604418325077881974066859, 4.74191036026273193172260542667, 5.70803714724611779299667061271, 6.60972766400120352272251827049, 7.02651647684643329954788268327, 7.84738728630060470723852843624, 8.754544724297870351345490955772