| L(s) = 1 | + 1.39·2-s − 3-s − 0.0594·4-s − 5-s − 1.39·6-s − 2.63·7-s − 2.86·8-s + 9-s − 1.39·10-s − 6.12·11-s + 0.0594·12-s − 4.25·13-s − 3.67·14-s + 15-s − 3.87·16-s − 0.114·17-s + 1.39·18-s − 4.89·19-s + 0.0594·20-s + 2.63·21-s − 8.53·22-s + 1.07·23-s + 2.86·24-s + 25-s − 5.93·26-s − 27-s + 0.156·28-s + ⋯ |
| L(s) = 1 | + 0.985·2-s − 0.577·3-s − 0.0297·4-s − 0.447·5-s − 0.568·6-s − 0.995·7-s − 1.01·8-s + 0.333·9-s − 0.440·10-s − 1.84·11-s + 0.0171·12-s − 1.18·13-s − 0.980·14-s + 0.258·15-s − 0.969·16-s − 0.0278·17-s + 0.328·18-s − 1.12·19-s + 0.0132·20-s + 0.574·21-s − 1.82·22-s + 0.224·23-s + 0.585·24-s + 0.200·25-s − 1.16·26-s − 0.192·27-s + 0.0295·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.002437408216\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.002437408216\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 401 | \( 1 - T \) |
| good | 2 | \( 1 - 1.39T + 2T^{2} \) |
| 7 | \( 1 + 2.63T + 7T^{2} \) |
| 11 | \( 1 + 6.12T + 11T^{2} \) |
| 13 | \( 1 + 4.25T + 13T^{2} \) |
| 17 | \( 1 + 0.114T + 17T^{2} \) |
| 19 | \( 1 + 4.89T + 19T^{2} \) |
| 23 | \( 1 - 1.07T + 23T^{2} \) |
| 29 | \( 1 + 7.76T + 29T^{2} \) |
| 31 | \( 1 - 0.623T + 31T^{2} \) |
| 37 | \( 1 - 1.43T + 37T^{2} \) |
| 41 | \( 1 + 2.55T + 41T^{2} \) |
| 43 | \( 1 + 5.42T + 43T^{2} \) |
| 47 | \( 1 + 2.89T + 47T^{2} \) |
| 53 | \( 1 - 0.713T + 53T^{2} \) |
| 59 | \( 1 + 12.4T + 59T^{2} \) |
| 61 | \( 1 + 0.909T + 61T^{2} \) |
| 67 | \( 1 + 14.2T + 67T^{2} \) |
| 71 | \( 1 - 11.4T + 71T^{2} \) |
| 73 | \( 1 + 11.2T + 73T^{2} \) |
| 79 | \( 1 - 8.76T + 79T^{2} \) |
| 83 | \( 1 - 6.20T + 83T^{2} \) |
| 89 | \( 1 + 2.01T + 89T^{2} \) |
| 97 | \( 1 - 10.5T + 97T^{2} \) |
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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
−7.84941177112382338475872862963, −7.29036521451072313379067123643, −6.39872424830647431357530215433, −5.85780469376989953575066896386, −4.95825915790911693622307290714, −4.75304511907428348448331113545, −3.69211568739542618545263826901, −3.00345808149422629555131645848, −2.22445192430190441973339887798, −0.02094679433124018722412287030,
0.02094679433124018722412287030, 2.22445192430190441973339887798, 3.00345808149422629555131645848, 3.69211568739542618545263826901, 4.75304511907428348448331113545, 4.95825915790911693622307290714, 5.85780469376989953575066896386, 6.39872424830647431357530215433, 7.29036521451072313379067123643, 7.84941177112382338475872862963
