L(s) = 1 | + 2-s + (0.707 + 0.707i)3-s + 4-s + (−2.22 − 0.220i)5-s + (0.707 + 0.707i)6-s + (−1.18 − 1.18i)7-s + 8-s + 1.00i·9-s + (−2.22 − 0.220i)10-s − 4.79i·11-s + (0.707 + 0.707i)12-s + 4.44·13-s + (−1.18 − 1.18i)14-s + (−1.41 − 1.72i)15-s + 16-s − 1.30i·17-s + ⋯ |
L(s) = 1 | + 0.707·2-s + (0.408 + 0.408i)3-s + 0.5·4-s + (−0.995 − 0.0987i)5-s + (0.288 + 0.288i)6-s + (−0.448 − 0.448i)7-s + 0.353·8-s + 0.333i·9-s + (−0.703 − 0.0698i)10-s − 1.44i·11-s + (0.204 + 0.204i)12-s + 1.23·13-s + (−0.316 − 0.316i)14-s + (−0.365 − 0.446i)15-s + 0.250·16-s − 0.315i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.812 + 0.583i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.812 + 0.583i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.386417949\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.386417949\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 + (-0.707 - 0.707i)T \) |
| 5 | \( 1 + (2.22 + 0.220i)T \) |
| 37 | \( 1 + (5.10 - 3.31i)T \) |
good | 7 | \( 1 + (1.18 + 1.18i)T + 7iT^{2} \) |
| 11 | \( 1 + 4.79iT - 11T^{2} \) |
| 13 | \( 1 - 4.44T + 13T^{2} \) |
| 17 | \( 1 + 1.30iT - 17T^{2} \) |
| 19 | \( 1 + (-4.46 + 4.46i)T - 19iT^{2} \) |
| 23 | \( 1 + 0.933T + 23T^{2} \) |
| 29 | \( 1 + (-3.66 - 3.66i)T + 29iT^{2} \) |
| 31 | \( 1 + (-4.54 + 4.54i)T - 31iT^{2} \) |
| 41 | \( 1 - 7.79iT - 41T^{2} \) |
| 43 | \( 1 - 2.02T + 43T^{2} \) |
| 47 | \( 1 + (6.81 + 6.81i)T + 47iT^{2} \) |
| 53 | \( 1 + (-5.08 + 5.08i)T - 53iT^{2} \) |
| 59 | \( 1 + (-3.87 + 3.87i)T - 59iT^{2} \) |
| 61 | \( 1 + (6.36 - 6.36i)T - 61iT^{2} \) |
| 67 | \( 1 + (0.639 - 0.639i)T - 67iT^{2} \) |
| 71 | \( 1 + 8.94T + 71T^{2} \) |
| 73 | \( 1 + (-1.06 - 1.06i)T + 73iT^{2} \) |
| 79 | \( 1 + (-2.21 + 2.21i)T - 79iT^{2} \) |
| 83 | \( 1 + (6.32 - 6.32i)T - 83iT^{2} \) |
| 89 | \( 1 + (3.11 + 3.11i)T + 89iT^{2} \) |
| 97 | \( 1 - 0.0235iT - 97T^{2} \) |
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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
−9.848457527672044406864805444026, −8.675506978742763295125515090850, −8.271721093768180164303214978134, −7.18199395855025248878221075281, −6.39579119963730650722039807609, −5.31865008375661415938202478974, −4.36163596855276284814146445513, −3.41330649728366092338201480933, −3.06145213869906322819166352316, −0.886188685250004521964924785743,
1.50414972042397588818383444782, 2.85245885038729717397868693337, 3.71303798845757439262254229276, 4.50835032421155614331805328992, 5.73929999561332765498442037970, 6.60590951792318709362816571698, 7.40256322863112182616524779544, 8.094116611269768992681768588748, 8.966879812022747877709678238897, 10.00853154078045075363692774365