| L(s) = 1 | − 1.40e5·2-s + 1.29e8·3-s + 1.11e10·4-s − 1.81e13·6-s + 4.00e13·7-s − 3.61e14·8-s + 1.11e16·9-s + 2.48e17·11-s + 1.44e18·12-s − 3.61e18·13-s − 5.62e18·14-s − 4.50e19·16-s + 5.03e19·17-s − 1.56e21·18-s − 1.19e21·19-s + 5.16e21·21-s − 3.48e22·22-s + 2.35e22·23-s − 4.67e22·24-s + 5.07e23·26-s + 7.17e23·27-s + 4.46e23·28-s − 2.45e24·29-s − 2.88e24·31-s + 9.43e24·32-s + 3.20e25·33-s − 7.07e24·34-s + ⋯ |
| L(s) = 1 | − 1.51·2-s + 1.73·3-s + 1.29·4-s − 2.62·6-s + 0.455·7-s − 0.454·8-s + 1.99·9-s + 1.62·11-s + 2.25·12-s − 1.50·13-s − 0.690·14-s − 0.610·16-s + 0.250·17-s − 3.03·18-s − 0.952·19-s + 0.788·21-s − 2.46·22-s + 0.800·23-s − 0.787·24-s + 2.28·26-s + 1.73·27-s + 0.591·28-s − 1.81·29-s − 0.712·31-s + 1.38·32-s + 2.82·33-s − 0.380·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(34-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+33/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(17)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{35}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| good | 2 | \( 1 + 1.40e5T + 8.58e9T^{2} \) |
| 3 | \( 1 - 1.29e8T + 5.55e15T^{2} \) |
| 7 | \( 1 - 4.00e13T + 7.73e27T^{2} \) |
| 11 | \( 1 - 2.48e17T + 2.32e34T^{2} \) |
| 13 | \( 1 + 3.61e18T + 5.75e36T^{2} \) |
| 17 | \( 1 - 5.03e19T + 4.02e40T^{2} \) |
| 19 | \( 1 + 1.19e21T + 1.58e42T^{2} \) |
| 23 | \( 1 - 2.35e22T + 8.65e44T^{2} \) |
| 29 | \( 1 + 2.45e24T + 1.81e48T^{2} \) |
| 31 | \( 1 + 2.88e24T + 1.64e49T^{2} \) |
| 37 | \( 1 + 3.58e25T + 5.63e51T^{2} \) |
| 41 | \( 1 - 6.61e25T + 1.66e53T^{2} \) |
| 43 | \( 1 + 1.70e27T + 8.02e53T^{2} \) |
| 47 | \( 1 + 4.53e27T + 1.51e55T^{2} \) |
| 53 | \( 1 + 1.11e28T + 7.96e56T^{2} \) |
| 59 | \( 1 - 1.06e29T + 2.74e58T^{2} \) |
| 61 | \( 1 - 9.42e28T + 8.23e58T^{2} \) |
| 67 | \( 1 + 2.59e30T + 1.82e60T^{2} \) |
| 71 | \( 1 + 4.27e30T + 1.23e61T^{2} \) |
| 73 | \( 1 + 2.23e30T + 3.08e61T^{2} \) |
| 79 | \( 1 + 2.21e31T + 4.18e62T^{2} \) |
| 83 | \( 1 + 2.60e31T + 2.13e63T^{2} \) |
| 89 | \( 1 - 1.65e32T + 2.13e64T^{2} \) |
| 97 | \( 1 - 1.37e32T + 3.65e65T^{2} \) |
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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
−10.06184476687862264869176269786, −9.260895293750154218772086487724, −8.666953940575348910831329389153, −7.61811180272267552596017327499, −6.90246253575137517541556820115, −4.51569073210976243618320952873, −3.27612131457892040087107770893, −1.94326724969270927053017899055, −1.55172079744186410150176179750, 0,
1.55172079744186410150176179750, 1.94326724969270927053017899055, 3.27612131457892040087107770893, 4.51569073210976243618320952873, 6.90246253575137517541556820115, 7.61811180272267552596017327499, 8.666953940575348910831329389153, 9.260895293750154218772086487724, 10.06184476687862264869176269786
