| L(s) = 1 | − 3.97e3·2-s + 2.45e4·3-s + 7.44e6·4-s + 1.19e8·5-s − 9.77e7·6-s + 3.74e9·8-s − 9.35e10·9-s − 4.74e11·10-s + 1.57e12·11-s + 1.82e11·12-s + 1.06e13·13-s + 2.93e12·15-s − 7.73e13·16-s − 1.15e14·17-s + 3.72e14·18-s + 3.11e14·19-s + 8.88e14·20-s − 6.25e15·22-s − 2.63e15·23-s + 9.21e13·24-s + 2.32e15·25-s − 4.22e16·26-s − 4.61e15·27-s + 1.24e17·29-s − 1.16e16·30-s + 1.82e17·31-s + 2.76e17·32-s + ⋯ |
| L(s) = 1 | − 1.37·2-s + 0.0800·3-s + 0.887·4-s + 1.09·5-s − 0.110·6-s + 0.154·8-s − 0.993·9-s − 1.50·10-s + 1.66·11-s + 0.0710·12-s + 1.64·13-s + 0.0875·15-s − 1.09·16-s − 0.813·17-s + 1.36·18-s + 0.613·19-s + 0.970·20-s − 2.28·22-s − 0.576·23-s + 0.0123·24-s + 0.195·25-s − 2.25·26-s − 0.159·27-s + 1.89·29-s − 0.120·30-s + 1.29·31-s + 1.35·32-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
| \(L(12)\) |
\(\approx\) |
\(1.785028539\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.785028539\) |
| \(L(\frac{25}{2})\) |
|
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 | \( 1 + 3.97e3T + 8.38e6T^{2} \) |
| 3 | \( 1 - 2.45e4T + 9.41e10T^{2} \) |
| 5 | \( 1 - 1.19e8T + 1.19e16T^{2} \) |
| 11 | \( 1 - 1.57e12T + 8.95e23T^{2} \) |
| 13 | \( 1 - 1.06e13T + 4.17e25T^{2} \) |
| 17 | \( 1 + 1.15e14T + 1.99e28T^{2} \) |
| 19 | \( 1 - 3.11e14T + 2.57e29T^{2} \) |
| 23 | \( 1 + 2.63e15T + 2.08e31T^{2} \) |
| 29 | \( 1 - 1.24e17T + 4.31e33T^{2} \) |
| 31 | \( 1 - 1.82e17T + 2.00e34T^{2} \) |
| 37 | \( 1 - 1.80e17T + 1.17e36T^{2} \) |
| 41 | \( 1 - 4.62e18T + 1.24e37T^{2} \) |
| 43 | \( 1 - 8.29e17T + 3.71e37T^{2} \) |
| 47 | \( 1 - 7.80e18T + 2.87e38T^{2} \) |
| 53 | \( 1 - 5.04e19T + 4.55e39T^{2} \) |
| 59 | \( 1 + 4.70e19T + 5.36e40T^{2} \) |
| 61 | \( 1 - 1.22e19T + 1.15e41T^{2} \) |
| 67 | \( 1 + 5.27e20T + 9.99e41T^{2} \) |
| 71 | \( 1 - 6.12e20T + 3.79e42T^{2} \) |
| 73 | \( 1 + 5.38e20T + 7.18e42T^{2} \) |
| 79 | \( 1 - 8.49e21T + 4.42e43T^{2} \) |
| 83 | \( 1 + 1.89e22T + 1.37e44T^{2} \) |
| 89 | \( 1 + 3.75e22T + 6.85e44T^{2} \) |
| 97 | \( 1 + 8.28e22T + 4.96e45T^{2} \) |
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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
−10.88181513263438152296754578301, −9.750238827338223763905064899007, −8.934024226491844542992928317705, −8.324349540597476583799382622427, −6.61915323002968746320694317751, −5.97463784441839910905895030900, −4.18398913885546422228466125499, −2.61685252234815214888248968758, −1.43962593563418085599096379383, −0.811240428626679348056491870891,
0.811240428626679348056491870891, 1.43962593563418085599096379383, 2.61685252234815214888248968758, 4.18398913885546422228466125499, 5.97463784441839910905895030900, 6.61915323002968746320694317751, 8.324349540597476583799382622427, 8.934024226491844542992928317705, 9.750238827338223763905064899007, 10.88181513263438152296754578301
