| L(s) = 1 | − 867.·2-s − 3.93e5·3-s − 7.63e6·4-s + 1.46e8·5-s + 3.41e8·6-s + 1.39e10·8-s + 6.03e10·9-s − 1.26e11·10-s − 1.21e12·11-s + 3.00e12·12-s + 9.16e12·13-s − 5.74e13·15-s + 5.19e13·16-s + 5.25e13·17-s − 5.23e13·18-s + 8.24e14·19-s − 1.11e15·20-s + 1.05e15·22-s + 2.00e15·23-s − 5.46e15·24-s + 9.46e15·25-s − 7.95e15·26-s + 1.32e16·27-s + 1.28e16·29-s + 4.98e16·30-s + 2.57e17·31-s − 1.61e17·32-s + ⋯ |
| L(s) = 1 | − 0.299·2-s − 1.28·3-s − 0.910·4-s + 1.33·5-s + 0.383·6-s + 0.572·8-s + 0.641·9-s − 0.401·10-s − 1.27·11-s + 1.16·12-s + 1.41·13-s − 1.71·15-s + 0.738·16-s + 0.371·17-s − 0.192·18-s + 1.62·19-s − 1.21·20-s + 0.383·22-s + 0.439·23-s − 0.733·24-s + 0.794·25-s − 0.425·26-s + 0.459·27-s + 0.195·29-s + 0.514·30-s + 1.82·31-s − 0.793·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(24-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+23/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(12)\) |
\(\approx\) |
\(1.367419021\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.367419021\) |
| \(L(\frac{25}{2})\) |
|
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 | \( 1 + 867.T + 8.38e6T^{2} \) |
| 3 | \( 1 + 3.93e5T + 9.41e10T^{2} \) |
| 5 | \( 1 - 1.46e8T + 1.19e16T^{2} \) |
| 11 | \( 1 + 1.21e12T + 8.95e23T^{2} \) |
| 13 | \( 1 - 9.16e12T + 4.17e25T^{2} \) |
| 17 | \( 1 - 5.25e13T + 1.99e28T^{2} \) |
| 19 | \( 1 - 8.24e14T + 2.57e29T^{2} \) |
| 23 | \( 1 - 2.00e15T + 2.08e31T^{2} \) |
| 29 | \( 1 - 1.28e16T + 4.31e33T^{2} \) |
| 31 | \( 1 - 2.57e17T + 2.00e34T^{2} \) |
| 37 | \( 1 - 5.69e16T + 1.17e36T^{2} \) |
| 41 | \( 1 - 4.76e17T + 1.24e37T^{2} \) |
| 43 | \( 1 + 7.55e17T + 3.71e37T^{2} \) |
| 47 | \( 1 - 2.74e19T + 2.87e38T^{2} \) |
| 53 | \( 1 + 9.15e19T + 4.55e39T^{2} \) |
| 59 | \( 1 + 7.57e19T + 5.36e40T^{2} \) |
| 61 | \( 1 + 5.95e20T + 1.15e41T^{2} \) |
| 67 | \( 1 + 1.28e21T + 9.99e41T^{2} \) |
| 71 | \( 1 + 6.22e20T + 3.79e42T^{2} \) |
| 73 | \( 1 - 1.49e21T + 7.18e42T^{2} \) |
| 79 | \( 1 - 1.10e22T + 4.42e43T^{2} \) |
| 83 | \( 1 - 1.30e22T + 1.37e44T^{2} \) |
| 89 | \( 1 + 2.41e21T + 6.85e44T^{2} \) |
| 97 | \( 1 + 1.34e22T + 4.96e45T^{2} \) |
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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
−10.83946045260235866832143147037, −10.16979384185040354768775782960, −9.168705663878371316292655845130, −7.86194858382226279215655581180, −6.20084624664325228841175091035, −5.52184648712778346808512397636, −4.77746937652312588453606969392, −3.02207419052949179656959850862, −1.32475184384035710067820703579, −0.66806850681568630023509371472,
0.66806850681568630023509371472, 1.32475184384035710067820703579, 3.02207419052949179656959850862, 4.77746937652312588453606969392, 5.52184648712778346808512397636, 6.20084624664325228841175091035, 7.86194858382226279215655581180, 9.168705663878371316292655845130, 10.16979384185040354768775782960, 10.83946045260235866832143147037
