L(s) = 1 | + i·2-s − 3-s − 4-s − i·5-s − i·6-s + 1.19i·7-s − i·8-s + 9-s + 10-s − 3.93i·11-s + 12-s − 1.19·14-s + i·15-s + 16-s − 1.74·17-s + i·18-s + ⋯ |
L(s) = 1 | + 0.707i·2-s − 0.577·3-s − 0.5·4-s − 0.447i·5-s − 0.408i·6-s + 0.452i·7-s − 0.353i·8-s + 0.333·9-s + 0.316·10-s − 1.18i·11-s + 0.288·12-s − 0.320·14-s + 0.258i·15-s + 0.250·16-s − 0.422·17-s + 0.235i·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0304i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.0304i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.2642260137\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2642260137\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - iT \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + iT \) |
| 13 | \( 1 \) |
good | 7 | \( 1 - 1.19iT - 7T^{2} \) |
| 11 | \( 1 + 3.93iT - 11T^{2} \) |
| 17 | \( 1 + 1.74T + 17T^{2} \) |
| 19 | \( 1 - 0.911iT - 19T^{2} \) |
| 23 | \( 1 + 1.24T + 23T^{2} \) |
| 29 | \( 1 + 3.47T + 29T^{2} \) |
| 31 | \( 1 + 2.15iT - 31T^{2} \) |
| 37 | \( 1 + 4.80iT - 37T^{2} \) |
| 41 | \( 1 - 0.198iT - 41T^{2} \) |
| 43 | \( 1 + 3.43T + 43T^{2} \) |
| 47 | \( 1 + 0.902iT - 47T^{2} \) |
| 53 | \( 1 + 6.91T + 53T^{2} \) |
| 59 | \( 1 - 10.2iT - 59T^{2} \) |
| 61 | \( 1 - 9.92T + 61T^{2} \) |
| 67 | \( 1 + 9.16iT - 67T^{2} \) |
| 71 | \( 1 - 11.0iT - 71T^{2} \) |
| 73 | \( 1 + 4.13iT - 73T^{2} \) |
| 79 | \( 1 - 2.73T + 79T^{2} \) |
| 83 | \( 1 - 15.5iT - 83T^{2} \) |
| 89 | \( 1 + 8.57iT - 89T^{2} \) |
| 97 | \( 1 - 16.9iT - 97T^{2} \) |
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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
−8.476060441460132339563649033942, −7.930091793254996476324182386941, −7.05687438714761721833710485341, −6.30083423530242615495459567291, −5.67480483297309462400098251448, −5.25242770349810533147357850405, −4.26752923216014626568798276477, −3.56334698325609932847959849054, −2.32534325939617105434455657022, −1.03087951186062140500447256184,
0.088394300558686218319412513190, 1.46210012705759021330957077697, 2.27432716196676775544224912232, 3.31203156791596538353386672196, 4.17991788459464686052686484190, 4.76647880556174834297057033258, 5.56685159382157903939288950518, 6.56320772888183143582877974786, 7.06633058715250098841436370778, 7.81231822606244726999125060242