| L(s) = 1 | + 2.84e3·2-s + 5.90e4·3-s + 5.99e6·4-s + 1.67e8·6-s − 3.63e8·7-s + 1.10e10·8-s + 3.48e9·9-s + 1.45e10·11-s + 3.53e11·12-s − 1.13e11·13-s − 1.03e12·14-s + 1.89e13·16-s + 8.58e12·17-s + 9.91e12·18-s − 2.92e13·19-s − 2.14e13·21-s + 4.14e13·22-s + 1.55e14·23-s + 6.53e14·24-s − 3.22e14·26-s + 2.05e14·27-s − 2.17e15·28-s + 2.40e15·29-s + 2.23e15·31-s + 3.06e16·32-s + 8.61e14·33-s + 2.44e16·34-s + ⋯ |
| L(s) = 1 | + 1.96·2-s + 0.577·3-s + 2.85·4-s + 1.13·6-s − 0.486·7-s + 3.64·8-s + 1/3·9-s + 0.169·11-s + 1.64·12-s − 0.228·13-s − 0.954·14-s + 4.30·16-s + 1.03·17-s + 0.654·18-s − 1.09·19-s − 0.280·21-s + 0.332·22-s + 0.784·23-s + 2.10·24-s − 0.447·26-s + 0.192·27-s − 1.38·28-s + 1.05·29-s + 0.490·31-s + 4.80·32-s + 0.0978·33-s + 2.02·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(11)\) |
\(\approx\) |
\(12.99929672\) |
| \(L(\frac12)\) |
\(\approx\) |
\(12.99929672\) |
| \(L(\frac{23}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - p^{10} T \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 - 711 p^{2} T + p^{21} T^{2} \) |
| 7 | \( 1 + 51900560 p T + p^{21} T^{2} \) |
| 11 | \( 1 - 1325621196 p T + p^{21} T^{2} \) |
| 13 | \( 1 + 113350790702 T + p^{21} T^{2} \) |
| 17 | \( 1 - 505258211646 p T + p^{21} T^{2} \) |
| 19 | \( 1 + 1536996803884 p T + p^{21} T^{2} \) |
| 23 | \( 1 - 155899214954280 T + p^{21} T^{2} \) |
| 29 | \( 1 - 2400788707090758 T + p^{21} T^{2} \) |
| 31 | \( 1 - 2239820676947000 T + p^{21} T^{2} \) |
| 37 | \( 1 - 30785069383298890 T + p^{21} T^{2} \) |
| 41 | \( 1 + 103207571041281030 T + p^{21} T^{2} \) |
| 43 | \( 1 - 165557270617488124 T + p^{21} T^{2} \) |
| 47 | \( 1 - 66587216226477408 T + p^{21} T^{2} \) |
| 53 | \( 1 + 435422766592881630 T + p^{21} T^{2} \) |
| 59 | \( 1 - 5534365798259081316 T + p^{21} T^{2} \) |
| 61 | \( 1 + 7176205164722961202 T + p^{21} T^{2} \) |
| 67 | \( 1 - 15755449453068299812 T + p^{21} T^{2} \) |
| 71 | \( 1 - 26457854874259376232 T + p^{21} T^{2} \) |
| 73 | \( 1 + 13471249335464801450 T + p^{21} T^{2} \) |
| 79 | \( 1 + 16886125085525986840 T + p^{21} T^{2} \) |
| 83 | \( 1 - \)\(17\!\cdots\!72\)\( T + p^{21} T^{2} \) |
| 89 | \( 1 + \)\(31\!\cdots\!86\)\( T + p^{21} T^{2} \) |
| 97 | \( 1 + \)\(94\!\cdots\!18\)\( T + p^{21} T^{2} \) |
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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
−10.96350147945993118311061478687, −9.898408452658526723912847904398, −8.131595523103801061780903601283, −6.97826532167550057196153541731, −6.17046180525845132829078743373, −5.00049138698976423635791447394, −4.05443291020460993441710828554, −3.12507262661524012162651351021, −2.38623740901262507887794601346, −1.15498866676382844994174510444,
1.15498866676382844994174510444, 2.38623740901262507887794601346, 3.12507262661524012162651351021, 4.05443291020460993441710828554, 5.00049138698976423635791447394, 6.17046180525845132829078743373, 6.97826532167550057196153541731, 8.131595523103801061780903601283, 9.898408452658526723912847904398, 10.96350147945993118311061478687
