| L(s) = 1 | + 3-s − 4.12i·5-s + 9-s + 4.83i·11-s + (−1.36 + 3.33i)13-s − 4.12i·15-s + 0.193·17-s + 0.814i·19-s + 1.54·23-s − 12.0·25-s + 27-s + 0.154·29-s − 4.84i·31-s + 4.83i·33-s − 0.110i·37-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.84i·5-s + 0.333·9-s + 1.45i·11-s + (−0.379 + 0.925i)13-s − 1.06i·15-s + 0.0468·17-s + 0.186i·19-s + 0.321·23-s − 2.40·25-s + 0.192·27-s + 0.0286·29-s − 0.870i·31-s + 0.840i·33-s − 0.0182i·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7644 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.379 - 0.925i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7644 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.379 - 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.541076212\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.541076212\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 \) |
| 13 | \( 1 + (1.36 - 3.33i)T \) |
| good | 5 | \( 1 + 4.12iT - 5T^{2} \) |
| 11 | \( 1 - 4.83iT - 11T^{2} \) |
| 17 | \( 1 - 0.193T + 17T^{2} \) |
| 19 | \( 1 - 0.814iT - 19T^{2} \) |
| 23 | \( 1 - 1.54T + 23T^{2} \) |
| 29 | \( 1 - 0.154T + 29T^{2} \) |
| 31 | \( 1 + 4.84iT - 31T^{2} \) |
| 37 | \( 1 + 0.110iT - 37T^{2} \) |
| 41 | \( 1 - 1.85iT - 41T^{2} \) |
| 43 | \( 1 + 1.19T + 43T^{2} \) |
| 47 | \( 1 - 9.78iT - 47T^{2} \) |
| 53 | \( 1 + 12.2T + 53T^{2} \) |
| 59 | \( 1 - 13.9iT - 59T^{2} \) |
| 61 | \( 1 - 5.27T + 61T^{2} \) |
| 67 | \( 1 - 11.7iT - 67T^{2} \) |
| 71 | \( 1 - 4.22iT - 71T^{2} \) |
| 73 | \( 1 - 3.21iT - 73T^{2} \) |
| 79 | \( 1 + 11.7T + 79T^{2} \) |
| 83 | \( 1 - 2.65iT - 83T^{2} \) |
| 89 | \( 1 + 4.95iT - 89T^{2} \) |
| 97 | \( 1 + 3.13iT - 97T^{2} \) |
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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
−8.001571993111288732128515111949, −7.51669874950681715314880800522, −6.74739210794853216431214242843, −5.77190059562847022108350603437, −4.98949311789638047669317884749, −4.36615090815564036644918081385, −4.09309691872165021058873204532, −2.67946763071070393654211026156, −1.81021712043676740808615071504, −1.18309819799261348522461099290,
0.32808769568800795428060910095, 1.85452646832937092606565010438, 2.85413716884956505010362349726, 3.19969211328476848647018135306, 3.73570600527814851147734355968, 5.01069282504015691068986950138, 5.78770802110894139412519320697, 6.53341667260352835636102834475, 6.97608036554747117717075946414, 7.82712578993672109220745024459
