L(s) = 1 | + 2.79·3-s − 2.61·5-s + 0.0133·7-s + 4.82·9-s + 0.228·11-s + 0.688·13-s − 7.32·15-s − 1.16·17-s − 0.379·19-s + 0.0373·21-s + 7.37·23-s + 1.86·25-s + 5.09·27-s − 4.16·29-s + 5.39·31-s + 0.639·33-s − 0.0350·35-s + 8.04·37-s + 1.92·39-s + 4.81·41-s − 4.53·43-s − 12.6·45-s + 5.68·47-s − 6.99·49-s − 3.25·51-s + 7.16·53-s − 0.598·55-s + ⋯ |
L(s) = 1 | + 1.61·3-s − 1.17·5-s + 0.00505·7-s + 1.60·9-s + 0.0688·11-s + 0.190·13-s − 1.89·15-s − 0.282·17-s − 0.0871·19-s + 0.00815·21-s + 1.53·23-s + 0.372·25-s + 0.981·27-s − 0.772·29-s + 0.968·31-s + 0.111·33-s − 0.00591·35-s + 1.32·37-s + 0.308·39-s + 0.752·41-s − 0.691·43-s − 1.88·45-s + 0.828·47-s − 0.999·49-s − 0.456·51-s + 0.984·53-s − 0.0807·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.028541691\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.028541691\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 751 | \( 1 + T \) |
good | 3 | \( 1 - 2.79T + 3T^{2} \) |
| 5 | \( 1 + 2.61T + 5T^{2} \) |
| 7 | \( 1 - 0.0133T + 7T^{2} \) |
| 11 | \( 1 - 0.228T + 11T^{2} \) |
| 13 | \( 1 - 0.688T + 13T^{2} \) |
| 17 | \( 1 + 1.16T + 17T^{2} \) |
| 19 | \( 1 + 0.379T + 19T^{2} \) |
| 23 | \( 1 - 7.37T + 23T^{2} \) |
| 29 | \( 1 + 4.16T + 29T^{2} \) |
| 31 | \( 1 - 5.39T + 31T^{2} \) |
| 37 | \( 1 - 8.04T + 37T^{2} \) |
| 41 | \( 1 - 4.81T + 41T^{2} \) |
| 43 | \( 1 + 4.53T + 43T^{2} \) |
| 47 | \( 1 - 5.68T + 47T^{2} \) |
| 53 | \( 1 - 7.16T + 53T^{2} \) |
| 59 | \( 1 + 11.9T + 59T^{2} \) |
| 61 | \( 1 - 3.14T + 61T^{2} \) |
| 67 | \( 1 - 12.4T + 67T^{2} \) |
| 71 | \( 1 + 11.7T + 71T^{2} \) |
| 73 | \( 1 + 2.77T + 73T^{2} \) |
| 79 | \( 1 - 13.8T + 79T^{2} \) |
| 83 | \( 1 - 6.29T + 83T^{2} \) |
| 89 | \( 1 - 8.77T + 89T^{2} \) |
| 97 | \( 1 + 5.10T + 97T^{2} \) |
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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
−8.073650844628417675096608482996, −7.57785902715300785852453914886, −7.02365970588039499458179463560, −6.10330817356846489420145839766, −4.85224875501287097628882568189, −4.23048113499422306356140636175, −3.51874073554025895220131101797, −2.94977562683412433478325547463, −2.08664306666273949705572739726, −0.853900460426784484379603272812,
0.853900460426784484379603272812, 2.08664306666273949705572739726, 2.94977562683412433478325547463, 3.51874073554025895220131101797, 4.23048113499422306356140636175, 4.85224875501287097628882568189, 6.10330817356846489420145839766, 7.02365970588039499458179463560, 7.57785902715300785852453914886, 8.073650844628417675096608482996