L(s) = 1 | + (1.52 + 2.58i)3-s + 7.48i·7-s + (−4.32 + 7.89i)9-s + 8.48i·11-s + 10i·13-s − 30.3·17-s + 26.9·19-s + (−19.3 + 11.4i)21-s + 9.17·23-s + (−26.9 + 0.905i)27-s − 26.8i·29-s − 8·31-s + (−21.9 + 12.9i)33-s + 15.9i·37-s + (−25.8 + 15.2i)39-s + ⋯ |
L(s) = 1 | + (0.509 + 0.860i)3-s + 1.06i·7-s + (−0.480 + 0.876i)9-s + 0.771i·11-s + 0.769i·13-s − 1.78·17-s + 1.41·19-s + (−0.920 + 0.545i)21-s + 0.398·23-s + (−0.999 + 0.0335i)27-s − 0.925i·29-s − 0.258·31-s + (−0.663 + 0.393i)33-s + 0.431i·37-s + (−0.661 + 0.392i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 + 0.0710i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.997 + 0.0710i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.495219645\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.495219645\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.52 - 2.58i)T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 - 7.48iT - 49T^{2} \) |
| 11 | \( 1 - 8.48iT - 121T^{2} \) |
| 13 | \( 1 - 10iT - 169T^{2} \) |
| 17 | \( 1 + 30.3T + 289T^{2} \) |
| 19 | \( 1 - 26.9T + 361T^{2} \) |
| 23 | \( 1 - 9.17T + 529T^{2} \) |
| 29 | \( 1 + 26.8iT - 841T^{2} \) |
| 31 | \( 1 + 8T + 961T^{2} \) |
| 37 | \( 1 - 15.9iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 47.3iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 14.4iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 45.8T + 2.20e3T^{2} \) |
| 53 | \( 1 - 30.3T + 2.80e3T^{2} \) |
| 59 | \( 1 - 24.0iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 53.9T + 3.72e3T^{2} \) |
| 67 | \( 1 - 110. iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 15.5iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 87.9iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 46.9T + 6.24e3T^{2} \) |
| 83 | \( 1 - 26.1T + 6.88e3T^{2} \) |
| 89 | \( 1 - 60.7iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 36.0iT - 9.40e3T^{2} \) |
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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
−9.739892809638056063244002175108, −9.136792856685344386991297358413, −8.688057914907731463478841708408, −7.60496943033833932359038188944, −6.68489886914610007086697846523, −5.56069404269890974753693201921, −4.77930044572617022271119484535, −3.96914768221884875704768132017, −2.72688692345037237981814499529, −1.97900601405488181742901218433,
0.40263235432078794796134316656, 1.43216593897770524434621632811, 2.85825799458431516945300067189, 3.59017832702613495604378399738, 4.81301818191082612966207515661, 5.95871856531592153829835901949, 6.84992628860909690069185661158, 7.44131337113060767344391127926, 8.251769709355844588178711875605, 8.996864661309946187799060625283