| L(s) = 1 | + 8.04e3·2-s − 9.02e5·3-s + 3.10e7·4-s + 9.26e8·5-s − 7.25e9·6-s − 1.97e10·8-s − 3.30e10·9-s + 7.44e12·10-s − 1.37e13·11-s − 2.80e13·12-s − 3.23e13·13-s − 8.35e14·15-s − 1.20e15·16-s + 1.03e15·17-s − 2.65e14·18-s − 1.17e16·19-s + 2.88e16·20-s − 1.10e17·22-s + 1.48e17·23-s + 1.78e16·24-s + 5.60e17·25-s − 2.60e17·26-s + 7.94e17·27-s + 7.97e17·29-s − 6.72e18·30-s + 3.05e18·31-s − 9.00e18·32-s + ⋯ |
| L(s) = 1 | + 1.38·2-s − 0.980·3-s + 0.926·4-s + 1.69·5-s − 1.36·6-s − 0.101·8-s − 0.0390·9-s + 2.35·10-s − 1.32·11-s − 0.908·12-s − 0.385·13-s − 1.66·15-s − 1.06·16-s + 0.430·17-s − 0.0541·18-s − 1.21·19-s + 1.57·20-s − 1.83·22-s + 1.40·23-s + 0.0996·24-s + 1.87·25-s − 0.535·26-s + 1.01·27-s + 0.418·29-s − 2.30·30-s + 0.697·31-s − 1.38·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(26-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+25/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(13)\) |
\(\approx\) |
\(3.739227443\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.739227443\) |
| \(L(\frac{27}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| good | 2 | \( 1 - 8.04e3T + 3.35e7T^{2} \) |
| 3 | \( 1 + 9.02e5T + 8.47e11T^{2} \) |
| 5 | \( 1 - 9.26e8T + 2.98e17T^{2} \) |
| 11 | \( 1 + 1.37e13T + 1.08e26T^{2} \) |
| 13 | \( 1 + 3.23e13T + 7.05e27T^{2} \) |
| 17 | \( 1 - 1.03e15T + 5.77e30T^{2} \) |
| 19 | \( 1 + 1.17e16T + 9.30e31T^{2} \) |
| 23 | \( 1 - 1.48e17T + 1.10e34T^{2} \) |
| 29 | \( 1 - 7.97e17T + 3.63e36T^{2} \) |
| 31 | \( 1 - 3.05e18T + 1.92e37T^{2} \) |
| 37 | \( 1 + 6.79e19T + 1.60e39T^{2} \) |
| 41 | \( 1 - 1.95e20T + 2.08e40T^{2} \) |
| 43 | \( 1 - 3.06e20T + 6.86e40T^{2} \) |
| 47 | \( 1 - 4.24e20T + 6.34e41T^{2} \) |
| 53 | \( 1 + 9.04e19T + 1.27e43T^{2} \) |
| 59 | \( 1 - 1.41e22T + 1.86e44T^{2} \) |
| 61 | \( 1 - 5.69e21T + 4.29e44T^{2} \) |
| 67 | \( 1 - 7.93e22T + 4.48e45T^{2} \) |
| 71 | \( 1 + 6.43e22T + 1.91e46T^{2} \) |
| 73 | \( 1 + 2.03e23T + 3.82e46T^{2} \) |
| 79 | \( 1 + 3.85e23T + 2.75e47T^{2} \) |
| 83 | \( 1 - 1.53e24T + 9.48e47T^{2} \) |
| 89 | \( 1 + 1.48e24T + 5.42e48T^{2} \) |
| 97 | \( 1 - 1.19e25T + 4.66e49T^{2} \) |
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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
−10.98128531563156051605692864243, −10.22626355153224809720033868562, −8.844235192587618633198183686630, −6.86127622055866855540350345633, −5.91654923571733784692167575044, −5.37949914530494211365008593038, −4.67750370569884890395621090618, −2.89559269583759734154881309306, −2.22234610505284009970056640522, −0.66590511843101454324118095669,
0.66590511843101454324118095669, 2.22234610505284009970056640522, 2.89559269583759734154881309306, 4.67750370569884890395621090618, 5.37949914530494211365008593038, 5.91654923571733784692167575044, 6.86127622055866855540350345633, 8.844235192587618633198183686630, 10.22626355153224809720033868562, 10.98128531563156051605692864243
