L(s) = 1 | − i·2-s − 4-s + (−1.41 + 2.23i)7-s + i·8-s + 1.41i·11-s − 5.39i·13-s + (2.23 + 1.41i)14-s + 16-s − 2.23·17-s + 1.30i·19-s + 1.41·22-s + i·23-s − 5.39·26-s + (1.41 − 2.23i)28-s − 9.24i·29-s + ⋯ |
L(s) = 1 | − 0.707i·2-s − 0.5·4-s + (−0.534 + 0.845i)7-s + 0.353i·8-s + 0.426i·11-s − 1.49i·13-s + (0.597 + 0.377i)14-s + 0.250·16-s − 0.542·17-s + 0.300i·19-s + 0.301·22-s + 0.208i·23-s − 1.05·26-s + (0.267 − 0.422i)28-s − 1.71i·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0515 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0515 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.290606612\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.290606612\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (1.41 - 2.23i)T \) |
good | 11 | \( 1 - 1.41iT - 11T^{2} \) |
| 13 | \( 1 + 5.39iT - 13T^{2} \) |
| 17 | \( 1 + 2.23T + 17T^{2} \) |
| 19 | \( 1 - 1.30iT - 19T^{2} \) |
| 23 | \( 1 - iT - 23T^{2} \) |
| 29 | \( 1 + 9.24iT - 29T^{2} \) |
| 31 | \( 1 - 8.56iT - 31T^{2} \) |
| 37 | \( 1 - 2.82T + 37T^{2} \) |
| 41 | \( 1 - 4.08T + 41T^{2} \) |
| 43 | \( 1 + 6.41T + 43T^{2} \) |
| 47 | \( 1 - 7.63T + 47T^{2} \) |
| 53 | \( 1 + 6.07iT - 53T^{2} \) |
| 59 | \( 1 - 11.7T + 59T^{2} \) |
| 61 | \( 1 - 5.39iT - 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 + 4.34iT - 71T^{2} \) |
| 73 | \( 1 + 5.01iT - 73T^{2} \) |
| 79 | \( 1 - 1.07T + 79T^{2} \) |
| 83 | \( 1 - 8.01T + 83T^{2} \) |
| 89 | \( 1 - 15.2T + 89T^{2} \) |
| 97 | \( 1 + 18.4iT - 97T^{2} \) |
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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
−8.557005554411402216436714162917, −7.993879467141414386437124289925, −7.01943396119104436894294424992, −6.03394829797157100097001945858, −5.44818946336630704585878688189, −4.56872395458930548383444545176, −3.53803757105314416894947795801, −2.79472200581479824323015195623, −1.97904439241271793969045367992, −0.52007116113276808299272177871,
0.890579118909995853677565278460, 2.31289236571616801977479521157, 3.59840946527648921598303312964, 4.21788174413398962775107168451, 5.01017298518742024131796216644, 6.07642567080402294545767337618, 6.68461351572152768737466923794, 7.20096260382696773941334219942, 8.001743519542606126943242589392, 8.988805803672968826334041151948