| L(s) = 1 | − 1.56i·3-s − i·7-s + 0.561·9-s − 6.12·11-s + 2i·13-s − 1.56i·17-s − 3.56·19-s − 1.56·21-s + 1.43i·23-s − 5.56i·27-s − 3.43·29-s − 9.12·31-s + 9.56i·33-s − 8.80i·37-s + 3.12·39-s + ⋯ |
| L(s) = 1 | − 0.901i·3-s − 0.377i·7-s + 0.187·9-s − 1.84·11-s + 0.554i·13-s − 0.378i·17-s − 0.817·19-s − 0.340·21-s + 0.299i·23-s − 1.07i·27-s − 0.638·29-s − 1.63·31-s + 1.66i·33-s − 1.44i·37-s + 0.500·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3259281401\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3259281401\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + iT \) |
| good | 3 | \( 1 + 1.56iT - 3T^{2} \) |
| 11 | \( 1 + 6.12T + 11T^{2} \) |
| 13 | \( 1 - 2iT - 13T^{2} \) |
| 17 | \( 1 + 1.56iT - 17T^{2} \) |
| 19 | \( 1 + 3.56T + 19T^{2} \) |
| 23 | \( 1 - 1.43iT - 23T^{2} \) |
| 29 | \( 1 + 3.43T + 29T^{2} \) |
| 31 | \( 1 + 9.12T + 31T^{2} \) |
| 37 | \( 1 + 8.80iT - 37T^{2} \) |
| 41 | \( 1 + 2.43T + 41T^{2} \) |
| 43 | \( 1 - 6.56iT - 43T^{2} \) |
| 47 | \( 1 - 8.24iT - 47T^{2} \) |
| 53 | \( 1 + 1.12iT - 53T^{2} \) |
| 59 | \( 1 + 11.3T + 59T^{2} \) |
| 61 | \( 1 - 11.1T + 61T^{2} \) |
| 67 | \( 1 + 7.87iT - 67T^{2} \) |
| 71 | \( 1 - 1.68T + 71T^{2} \) |
| 73 | \( 1 - 6.43iT - 73T^{2} \) |
| 79 | \( 1 - 5.68T + 79T^{2} \) |
| 83 | \( 1 + 1.31iT - 83T^{2} \) |
| 89 | \( 1 + 9.80T + 89T^{2} \) |
| 97 | \( 1 + 6iT - 97T^{2} \) |
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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
−9.081162752673009206428389654400, −7.958717293194539767873055845748, −7.55280305967758657100233381963, −6.83488907603964730613411195985, −5.85005395599561985619280843546, −4.95880589178186559231077555954, −3.91982692896788961672814128614, −2.60554095707743777150431627647, −1.71510285824784588129406347453, −0.12143691650617283800001316686,
2.03334467768135975053386425005, 3.12731312007437915747506060590, 4.08372290730380335181854380122, 5.15063957064894969698554384189, 5.52212033311335163036353834290, 6.79253430005078459046597179521, 7.76356082406297453785215464323, 8.454093945827793902222504359180, 9.287417716803989372260488826399, 10.29400094772509713632163057182
