| L(s) = 1 | − 1.08·3-s − 5-s + 2.80·7-s − 1.81·9-s + 2.75·11-s + 4.91·13-s + 1.08·15-s + 2.79·17-s + 1.91·19-s − 3.05·21-s − 2.47·23-s + 25-s + 5.23·27-s + 3.90·29-s + 4.75·31-s − 2.99·33-s − 2.80·35-s + 5.10·37-s − 5.34·39-s + 6.19·41-s − 10.6·43-s + 1.81·45-s + 5.90·47-s + 0.885·49-s − 3.04·51-s − 8.75·53-s − 2.75·55-s + ⋯ |
| L(s) = 1 | − 0.628·3-s − 0.447·5-s + 1.06·7-s − 0.605·9-s + 0.830·11-s + 1.36·13-s + 0.280·15-s + 0.678·17-s + 0.438·19-s − 0.666·21-s − 0.516·23-s + 0.200·25-s + 1.00·27-s + 0.724·29-s + 0.854·31-s − 0.521·33-s − 0.474·35-s + 0.839·37-s − 0.855·39-s + 0.967·41-s − 1.62·43-s + 0.270·45-s + 0.860·47-s + 0.126·49-s − 0.426·51-s − 1.20·53-s − 0.371·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.900074719\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.900074719\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 151 | \( 1 - T \) |
| good | 3 | \( 1 + 1.08T + 3T^{2} \) |
| 7 | \( 1 - 2.80T + 7T^{2} \) |
| 11 | \( 1 - 2.75T + 11T^{2} \) |
| 13 | \( 1 - 4.91T + 13T^{2} \) |
| 17 | \( 1 - 2.79T + 17T^{2} \) |
| 19 | \( 1 - 1.91T + 19T^{2} \) |
| 23 | \( 1 + 2.47T + 23T^{2} \) |
| 29 | \( 1 - 3.90T + 29T^{2} \) |
| 31 | \( 1 - 4.75T + 31T^{2} \) |
| 37 | \( 1 - 5.10T + 37T^{2} \) |
| 41 | \( 1 - 6.19T + 41T^{2} \) |
| 43 | \( 1 + 10.6T + 43T^{2} \) |
| 47 | \( 1 - 5.90T + 47T^{2} \) |
| 53 | \( 1 + 8.75T + 53T^{2} \) |
| 59 | \( 1 + 0.0340T + 59T^{2} \) |
| 61 | \( 1 + 4.60T + 61T^{2} \) |
| 67 | \( 1 + 3.15T + 67T^{2} \) |
| 71 | \( 1 + 5.26T + 71T^{2} \) |
| 73 | \( 1 + 5.59T + 73T^{2} \) |
| 79 | \( 1 + 3.41T + 79T^{2} \) |
| 83 | \( 1 - 13.0T + 83T^{2} \) |
| 89 | \( 1 - 17.9T + 89T^{2} \) |
| 97 | \( 1 + 0.569T + 97T^{2} \) |
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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
−8.097821419257394942005996078479, −7.51092890515056709840798305714, −6.32205521979085784847911920163, −6.14894377884110824512724437509, −5.16207393951356785550085660729, −4.53734610786352040350645513486, −3.72059682384719033574319874252, −2.89811753679477326008149850521, −1.54621353299056100598728099585, −0.825225300189948389669420287825,
0.825225300189948389669420287825, 1.54621353299056100598728099585, 2.89811753679477326008149850521, 3.72059682384719033574319874252, 4.53734610786352040350645513486, 5.16207393951356785550085660729, 6.14894377884110824512724437509, 6.32205521979085784847911920163, 7.51092890515056709840798305714, 8.097821419257394942005996078479
