| L(s) = 1 | − i·2-s + i·3-s − 4-s + (−0.806 + 2.08i)5-s + 6-s + i·8-s − 9-s + (2.08 + 0.806i)10-s − 4.78·11-s − i·12-s − 3.17i·13-s + (−2.08 − 0.806i)15-s + 16-s − 5.22i·17-s + i·18-s + 3.17·19-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s + 0.577i·3-s − 0.5·4-s + (−0.360 + 0.932i)5-s + 0.408·6-s + 0.353i·8-s − 0.333·9-s + (0.659 + 0.255i)10-s − 1.44·11-s − 0.288i·12-s − 0.879i·13-s + (−0.538 − 0.208i)15-s + 0.250·16-s − 1.26i·17-s + 0.235i·18-s + 0.727·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.360 + 0.932i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.360 + 0.932i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7004560077\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7004560077\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - iT \) |
| 5 | \( 1 + (0.806 - 2.08i)T \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + 4.78T + 11T^{2} \) |
| 13 | \( 1 + 3.17iT - 13T^{2} \) |
| 17 | \( 1 + 5.22iT - 17T^{2} \) |
| 19 | \( 1 - 3.17T + 19T^{2} \) |
| 23 | \( 1 - 7.17iT - 23T^{2} \) |
| 29 | \( 1 + 2.38T + 29T^{2} \) |
| 31 | \( 1 - 4.17T + 31T^{2} \) |
| 37 | \( 1 + 7.17iT - 37T^{2} \) |
| 41 | \( 1 + 2.05T + 41T^{2} \) |
| 43 | \( 1 + 2iT - 43T^{2} \) |
| 47 | \( 1 + 11.5iT - 47T^{2} \) |
| 53 | \( 1 + 8.11iT - 53T^{2} \) |
| 59 | \( 1 - 10.7T + 59T^{2} \) |
| 61 | \( 1 - 7.11T + 61T^{2} \) |
| 67 | \( 1 + 9.56iT - 67T^{2} \) |
| 71 | \( 1 - 6T + 71T^{2} \) |
| 73 | \( 1 - 4iT - 73T^{2} \) |
| 79 | \( 1 + 12.1T + 79T^{2} \) |
| 83 | \( 1 + 3.39iT - 83T^{2} \) |
| 89 | \( 1 + 9.56T + 89T^{2} \) |
| 97 | \( 1 - 1.27iT - 97T^{2} \) |
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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
−9.726195447399978939445185130500, −8.482893529873152341066407298543, −7.70297798402267478415598347897, −7.05789276295899408358927060939, −5.49202468229797116105858893132, −5.21221671930387086648674689568, −3.81510560408775911598319439865, −3.11594778067769678181324130957, −2.36621599225756956351554951738, −0.30251969546203392118981620802,
1.21748140205341073717931451469, 2.64097155902423886063811277505, 4.07697810550151471510627519791, 4.85854121421563609386384187848, 5.69129310351816057148540611943, 6.52866473747056601211976152371, 7.44406749620301637277630558705, 8.238154472858614558785564880910, 8.509961588262867331266455916778, 9.569224775113792745576300707619
