| L(s) = 1 | − 9.64e3·2-s + 1.23e6·3-s + 5.95e7·4-s + 2.52e8·5-s − 1.18e10·6-s − 2.50e11·8-s + 6.67e11·9-s − 2.43e12·10-s − 1.14e13·11-s + 7.32e13·12-s + 1.40e14·13-s + 3.10e14·15-s + 4.21e14·16-s − 1.07e14·17-s − 6.44e15·18-s + 1.59e16·19-s + 1.50e16·20-s + 1.10e17·22-s − 2.21e15·23-s − 3.08e17·24-s − 2.34e17·25-s − 1.35e18·26-s − 2.21e17·27-s − 9.32e16·29-s − 2.99e18·30-s + 3.74e18·31-s + 4.34e18·32-s + ⋯ |
| L(s) = 1 | − 1.66·2-s + 1.33·3-s + 1.77·4-s + 0.462·5-s − 2.22·6-s − 1.29·8-s + 0.787·9-s − 0.769·10-s − 1.09·11-s + 2.37·12-s + 1.67·13-s + 0.617·15-s + 0.374·16-s − 0.0447·17-s − 1.31·18-s + 1.65·19-s + 0.820·20-s + 1.83·22-s − 0.0210·23-s − 1.72·24-s − 0.786·25-s − 2.79·26-s − 0.283·27-s − 0.0489·29-s − 1.02·30-s + 0.852·31-s + 0.666·32-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
| \(L(13)\) |
\(\approx\) |
\(1.975045142\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.975045142\) |
| \(L(\frac{27}{2})\) |
|
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 | \( 1 + 9.64e3T + 3.35e7T^{2} \) |
| 3 | \( 1 - 1.23e6T + 8.47e11T^{2} \) |
| 5 | \( 1 - 2.52e8T + 2.98e17T^{2} \) |
| 11 | \( 1 + 1.14e13T + 1.08e26T^{2} \) |
| 13 | \( 1 - 1.40e14T + 7.05e27T^{2} \) |
| 17 | \( 1 + 1.07e14T + 5.77e30T^{2} \) |
| 19 | \( 1 - 1.59e16T + 9.30e31T^{2} \) |
| 23 | \( 1 + 2.21e15T + 1.10e34T^{2} \) |
| 29 | \( 1 + 9.32e16T + 3.63e36T^{2} \) |
| 31 | \( 1 - 3.74e18T + 1.92e37T^{2} \) |
| 37 | \( 1 + 6.76e19T + 1.60e39T^{2} \) |
| 41 | \( 1 - 1.18e20T + 2.08e40T^{2} \) |
| 43 | \( 1 + 2.95e20T + 6.86e40T^{2} \) |
| 47 | \( 1 - 8.34e20T + 6.34e41T^{2} \) |
| 53 | \( 1 - 2.97e21T + 1.27e43T^{2} \) |
| 59 | \( 1 - 1.93e22T + 1.86e44T^{2} \) |
| 61 | \( 1 - 2.53e21T + 4.29e44T^{2} \) |
| 67 | \( 1 - 9.75e22T + 4.48e45T^{2} \) |
| 71 | \( 1 - 2.43e23T + 1.91e46T^{2} \) |
| 73 | \( 1 + 1.75e23T + 3.82e46T^{2} \) |
| 79 | \( 1 + 7.09e23T + 2.75e47T^{2} \) |
| 83 | \( 1 + 8.81e23T + 9.48e47T^{2} \) |
| 89 | \( 1 - 2.12e24T + 5.42e48T^{2} \) |
| 97 | \( 1 - 6.70e23T + 4.66e49T^{2} \) |
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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
−10.36022174831996941483243510105, −9.592500237830438834479780638475, −8.627668805927127155833491229533, −8.084320319044310704692722310609, −7.08766997341505183000498730181, −5.61967714233769422367217445572, −3.57883066712030015399921721963, −2.57751873619972670533580843727, −1.69021120016796351871431071755, −0.73181606246207173721780291974,
0.73181606246207173721780291974, 1.69021120016796351871431071755, 2.57751873619972670533580843727, 3.57883066712030015399921721963, 5.61967714233769422367217445572, 7.08766997341505183000498730181, 8.084320319044310704692722310609, 8.627668805927127155833491229533, 9.592500237830438834479780638475, 10.36022174831996941483243510105
