L(s) = 1 | + (−2.03 + 0.918i)5-s + (1.31 − 2.29i)7-s + (−5.56 + 3.21i)11-s + (1.53 − 1.53i)13-s + (3.67 + 0.985i)17-s + (−1.19 − 0.687i)19-s + (8.15 − 2.18i)23-s + (3.31 − 3.74i)25-s + 5.95·29-s + (4.97 + 8.61i)31-s + (−0.566 + 5.88i)35-s + (−2.39 + 0.642i)37-s − 1.18i·41-s + (3.28 − 3.28i)43-s + (−0.655 − 2.44i)47-s + ⋯ |
L(s) = 1 | + (−0.911 + 0.410i)5-s + (0.495 − 0.868i)7-s + (−1.67 + 0.967i)11-s + (0.425 − 0.425i)13-s + (0.892 + 0.239i)17-s + (−0.273 − 0.157i)19-s + (1.70 − 0.455i)23-s + (0.662 − 0.748i)25-s + 1.10·29-s + (0.893 + 1.54i)31-s + (−0.0957 + 0.995i)35-s + (−0.394 + 0.105i)37-s − 0.184i·41-s + (0.500 − 0.500i)43-s + (−0.0955 − 0.356i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 - 0.0892i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.996 - 0.0892i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.362343807\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.362343807\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (2.03 - 0.918i)T \) |
| 7 | \( 1 + (-1.31 + 2.29i)T \) |
good | 11 | \( 1 + (5.56 - 3.21i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-1.53 + 1.53i)T - 13iT^{2} \) |
| 17 | \( 1 + (-3.67 - 0.985i)T + (14.7 + 8.5i)T^{2} \) |
| 19 | \( 1 + (1.19 + 0.687i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-8.15 + 2.18i)T + (19.9 - 11.5i)T^{2} \) |
| 29 | \( 1 - 5.95T + 29T^{2} \) |
| 31 | \( 1 + (-4.97 - 8.61i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (2.39 - 0.642i)T + (32.0 - 18.5i)T^{2} \) |
| 41 | \( 1 + 1.18iT - 41T^{2} \) |
| 43 | \( 1 + (-3.28 + 3.28i)T - 43iT^{2} \) |
| 47 | \( 1 + (0.655 + 2.44i)T + (-40.7 + 23.5i)T^{2} \) |
| 53 | \( 1 + (0.482 - 1.80i)T + (-45.8 - 26.5i)T^{2} \) |
| 59 | \( 1 + (-3.27 - 5.67i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-4.09 + 7.08i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (2.49 - 9.29i)T + (-58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 + 8.06iT - 71T^{2} \) |
| 73 | \( 1 + (-12.9 - 3.47i)T + (63.2 + 36.5i)T^{2} \) |
| 79 | \( 1 + (-8.00 - 4.62i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-5.11 - 5.11i)T + 83iT^{2} \) |
| 89 | \( 1 + (-3.97 + 6.88i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (0.537 + 0.537i)T + 97iT^{2} \) |
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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
−10.05037997266485809749916200761, −8.568331787133148044864191351530, −8.034895498910777179223242935564, −7.29043571340036208609186863540, −6.71217416020368252679677235014, −5.16583142240397075354021879226, −4.68757640912187080601612850085, −3.51172182494211510520113732258, −2.62528436815574587432725026986, −0.891471111330452992696372843776,
0.857487632386513337474819128832, 2.57240577514168079093268254107, 3.41332487800653919446180227876, 4.74363444708460077745015411681, 5.30537904708837309742400952703, 6.23030556488643862689183999952, 7.53462444851797919348815664427, 8.118354534800904411727267898874, 8.642444332265744593701552138189, 9.545356241093894940407884841260