| L(s) = 1 | + 1.88·2-s + 3-s + 1.53·4-s − 2.44·5-s + 1.88·6-s − 0.867·8-s + 9-s − 4.59·10-s + 3.12·11-s + 1.53·12-s − 3.06·13-s − 2.44·15-s − 4.70·16-s + 4.97·17-s + 1.88·18-s + 3.78·19-s − 3.75·20-s + 5.87·22-s − 8.35·23-s − 0.867·24-s + 0.957·25-s − 5.76·26-s + 27-s − 8.89·29-s − 4.59·30-s + 3.12·31-s − 7.12·32-s + ⋯ |
| L(s) = 1 | + 1.33·2-s + 0.577·3-s + 0.769·4-s − 1.09·5-s + 0.768·6-s − 0.306·8-s + 0.333·9-s − 1.45·10-s + 0.942·11-s + 0.444·12-s − 0.849·13-s − 0.630·15-s − 1.17·16-s + 1.20·17-s + 0.443·18-s + 0.867·19-s − 0.839·20-s + 1.25·22-s − 1.74·23-s − 0.177·24-s + 0.191·25-s − 1.13·26-s + 0.192·27-s − 1.65·29-s − 0.838·30-s + 0.560·31-s − 1.25·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7203 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7203 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - 1.88T + 2T^{2} \) |
| 5 | \( 1 + 2.44T + 5T^{2} \) |
| 11 | \( 1 - 3.12T + 11T^{2} \) |
| 13 | \( 1 + 3.06T + 13T^{2} \) |
| 17 | \( 1 - 4.97T + 17T^{2} \) |
| 19 | \( 1 - 3.78T + 19T^{2} \) |
| 23 | \( 1 + 8.35T + 23T^{2} \) |
| 29 | \( 1 + 8.89T + 29T^{2} \) |
| 31 | \( 1 - 3.12T + 31T^{2} \) |
| 37 | \( 1 + 7.30T + 37T^{2} \) |
| 41 | \( 1 - 4.34T + 41T^{2} \) |
| 43 | \( 1 - 4.14T + 43T^{2} \) |
| 47 | \( 1 - 1.75T + 47T^{2} \) |
| 53 | \( 1 + 0.616T + 53T^{2} \) |
| 59 | \( 1 - 2.96T + 59T^{2} \) |
| 61 | \( 1 + 8.12T + 61T^{2} \) |
| 67 | \( 1 + 7.14T + 67T^{2} \) |
| 71 | \( 1 + 10.0T + 71T^{2} \) |
| 73 | \( 1 + 2.05T + 73T^{2} \) |
| 79 | \( 1 - 4.68T + 79T^{2} \) |
| 83 | \( 1 + 12.2T + 83T^{2} \) |
| 89 | \( 1 + 2.12T + 89T^{2} \) |
| 97 | \( 1 + 19.4T + 97T^{2} \) |
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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
−7.54710354895963783425356794071, −6.92558638588412810816639812860, −5.91641137031946525789401062599, −5.42544054802790467470624436488, −4.41172471308159584517974123552, −3.96338441575440485832200363473, −3.44909836483607747542789557197, −2.68896025982882497420802133744, −1.56626280634608182271473258170, 0,
1.56626280634608182271473258170, 2.68896025982882497420802133744, 3.44909836483607747542789557197, 3.96338441575440485832200363473, 4.41172471308159584517974123552, 5.42544054802790467470624436488, 5.91641137031946525789401062599, 6.92558638588412810816639812860, 7.54710354895963783425356794071
