L(s) = 1 | + (1.68 − 0.418i)3-s + (1.37 − 2.37i)5-s + (−2.60 − 0.463i)7-s + (2.65 − 1.40i)9-s + (−0.362 + 0.209i)11-s + (1.32 − 0.765i)13-s + (1.31 − 4.56i)15-s + (−1.95 + 3.38i)17-s + (−5.11 + 2.95i)19-s + (−4.57 + 0.309i)21-s + (7.72 + 4.46i)23-s + (−1.26 − 2.18i)25-s + (3.86 − 3.47i)27-s + (6.00 + 3.46i)29-s − 3.52i·31-s + ⋯ |
L(s) = 1 | + (0.970 − 0.241i)3-s + (0.613 − 1.06i)5-s + (−0.984 − 0.175i)7-s + (0.883 − 0.468i)9-s + (−0.109 + 0.0630i)11-s + (0.367 − 0.212i)13-s + (0.338 − 1.17i)15-s + (−0.473 + 0.820i)17-s + (−1.17 + 0.678i)19-s + (−0.997 + 0.0674i)21-s + (1.61 + 0.930i)23-s + (−0.252 − 0.437i)25-s + (0.744 − 0.667i)27-s + (1.11 + 0.643i)29-s − 0.633i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.708 + 0.705i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.708 + 0.705i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.56407 - 0.646336i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.56407 - 0.646336i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.68 + 0.418i)T \) |
| 7 | \( 1 + (2.60 + 0.463i)T \) |
good | 5 | \( 1 + (-1.37 + 2.37i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (0.362 - 0.209i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-1.32 + 0.765i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (1.95 - 3.38i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (5.11 - 2.95i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-7.72 - 4.46i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-6.00 - 3.46i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + 3.52iT - 31T^{2} \) |
| 37 | \( 1 + (4.54 + 7.87i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-1.06 - 1.84i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (5.77 - 10.0i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + 1.77T + 47T^{2} \) |
| 53 | \( 1 + (3.39 + 1.96i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + 4.05T + 59T^{2} \) |
| 61 | \( 1 - 1.86iT - 61T^{2} \) |
| 67 | \( 1 + 12.7T + 67T^{2} \) |
| 71 | \( 1 - 8.51iT - 71T^{2} \) |
| 73 | \( 1 + (-1.65 - 0.952i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + 0.867T + 79T^{2} \) |
| 83 | \( 1 + (-3.45 + 5.99i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (4.88 + 8.46i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-0.200 - 0.115i)T + (48.5 + 84.0i)T^{2} \) |
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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
−12.45145428201346159578995786956, −10.74833618418184855751385262731, −9.746961312844519578626616012882, −8.983715532723879585332202427620, −8.295765990573993938024289262390, −6.96026953200161593465051997867, −5.92001493029038132943217514673, −4.43488745180310512407192383180, −3.15856091012414130330540439182, −1.53608820074980497259687498107,
2.46938430979405413340116735714, 3.18945251692596944068788348385, 4.73996670756605850971546548876, 6.56100084170261127547120122185, 6.89187352684808854844654052275, 8.540298379985326596474816206968, 9.237429504493670841909767062477, 10.28189000953891910447471013815, 10.80845243902836479657770859197, 12.33428560675635372238845333900