| L(s) = 1 | − 2.73·2-s − 1.54·3-s + 5.45·4-s − 5-s + 4.22·6-s + 1.26·7-s − 9.43·8-s − 0.607·9-s + 2.73·10-s − 0.417·11-s − 8.44·12-s − 5.66·13-s − 3.45·14-s + 1.54·15-s + 14.8·16-s + 1.65·18-s − 0.134·19-s − 5.45·20-s − 1.95·21-s + 1.14·22-s − 6.81·23-s + 14.5·24-s + 25-s + 15.4·26-s + 5.57·27-s + 6.89·28-s − 0.329·29-s + ⋯ |
| L(s) = 1 | − 1.93·2-s − 0.893·3-s + 2.72·4-s − 0.447·5-s + 1.72·6-s + 0.477·7-s − 3.33·8-s − 0.202·9-s + 0.863·10-s − 0.126·11-s − 2.43·12-s − 1.57·13-s − 0.922·14-s + 0.399·15-s + 3.71·16-s + 0.390·18-s − 0.0307·19-s − 1.22·20-s − 0.426·21-s + 0.243·22-s − 1.42·23-s + 2.98·24-s + 0.200·25-s + 3.03·26-s + 1.07·27-s + 1.30·28-s − 0.0611·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1445 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1445 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2279648870\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2279648870\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + T \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 + 2.73T + 2T^{2} \) |
| 3 | \( 1 + 1.54T + 3T^{2} \) |
| 7 | \( 1 - 1.26T + 7T^{2} \) |
| 11 | \( 1 + 0.417T + 11T^{2} \) |
| 13 | \( 1 + 5.66T + 13T^{2} \) |
| 19 | \( 1 + 0.134T + 19T^{2} \) |
| 23 | \( 1 + 6.81T + 23T^{2} \) |
| 29 | \( 1 + 0.329T + 29T^{2} \) |
| 31 | \( 1 - 0.605T + 31T^{2} \) |
| 37 | \( 1 - 8.83T + 37T^{2} \) |
| 41 | \( 1 + 1.63T + 41T^{2} \) |
| 43 | \( 1 + 1.99T + 43T^{2} \) |
| 47 | \( 1 + 5.06T + 47T^{2} \) |
| 53 | \( 1 + 1.54T + 53T^{2} \) |
| 59 | \( 1 + 1.41T + 59T^{2} \) |
| 61 | \( 1 - 7.80T + 61T^{2} \) |
| 67 | \( 1 + 12.1T + 67T^{2} \) |
| 71 | \( 1 + 4.82T + 71T^{2} \) |
| 73 | \( 1 - 2.90T + 73T^{2} \) |
| 79 | \( 1 + 4.33T + 79T^{2} \) |
| 83 | \( 1 + 4.39T + 83T^{2} \) |
| 89 | \( 1 - 4.98T + 89T^{2} \) |
| 97 | \( 1 - 0.635T + 97T^{2} \) |
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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
−9.689194009864277115083844487441, −8.656710769967298387568444439144, −7.957674393740685354688954462776, −7.41713075274566093746199162975, −6.52114505761563497136681418279, −5.74377060518100903041760781725, −4.65253713733975887334258455308, −2.94178219057648883423891267341, −1.91103652810641360559638686839, −0.45429506631537195866721982828,
0.45429506631537195866721982828, 1.91103652810641360559638686839, 2.94178219057648883423891267341, 4.65253713733975887334258455308, 5.74377060518100903041760781725, 6.52114505761563497136681418279, 7.41713075274566093746199162975, 7.957674393740685354688954462776, 8.656710769967298387568444439144, 9.689194009864277115083844487441
