| L(s) = 1 | + 1.60e6·5-s − 9.41e6·7-s + 1.86e8·11-s − 2.62e9·13-s − 4.37e10·17-s − 9.65e10·19-s − 2.90e11·23-s + 1.82e12·25-s − 1.39e12·29-s + 7.64e12·31-s − 1.51e13·35-s − 3.33e13·37-s + 1.20e13·41-s − 7.55e11·43-s + 2.80e14·47-s − 1.43e14·49-s − 4.60e14·53-s + 3.00e14·55-s − 1.07e15·59-s − 1.98e15·61-s − 4.22e15·65-s + 4.85e15·67-s − 2.70e15·71-s − 5.00e15·73-s − 1.76e15·77-s − 9.77e15·79-s − 1.71e16·83-s + ⋯ |
| L(s) = 1 | + 1.84·5-s − 0.617·7-s + 0.262·11-s − 0.892·13-s − 1.52·17-s − 1.30·19-s − 0.774·23-s + 2.39·25-s − 0.519·29-s + 1.61·31-s − 1.13·35-s − 1.56·37-s + 0.235·41-s − 0.00985·43-s + 1.71·47-s − 0.618·49-s − 1.01·53-s + 0.484·55-s − 0.956·59-s − 1.32·61-s − 1.64·65-s + 1.45·67-s − 0.497·71-s − 0.725·73-s − 0.162·77-s − 0.724·79-s − 0.833·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(18-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s+17/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(9)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{19}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 321786 p T + p^{17} T^{2} \) |
| 7 | \( 1 + 1345312 p T + p^{17} T^{2} \) |
| 11 | \( 1 - 186910524 T + p^{17} T^{2} \) |
| 13 | \( 1 + 201957130 p T + p^{17} T^{2} \) |
| 17 | \( 1 + 43782311106 T + p^{17} T^{2} \) |
| 19 | \( 1 + 96594985540 T + p^{17} T^{2} \) |
| 23 | \( 1 + 290867937336 T + p^{17} T^{2} \) |
| 29 | \( 1 + 1398617429094 T + p^{17} T^{2} \) |
| 31 | \( 1 - 7647898359464 T + p^{17} T^{2} \) |
| 37 | \( 1 + 33369516616762 T + p^{17} T^{2} \) |
| 41 | \( 1 - 12032733393990 T + p^{17} T^{2} \) |
| 43 | \( 1 + 755092495804 T + p^{17} T^{2} \) |
| 47 | \( 1 - 280540358127936 T + p^{17} T^{2} \) |
| 53 | \( 1 + 460570203615582 T + p^{17} T^{2} \) |
| 59 | \( 1 + 1078467799153284 T + p^{17} T^{2} \) |
| 61 | \( 1 + 1980778975313218 T + p^{17} T^{2} \) |
| 67 | \( 1 - 4850190377589884 T + p^{17} T^{2} \) |
| 71 | \( 1 + 2707574704052040 T + p^{17} T^{2} \) |
| 73 | \( 1 + 5002264428090742 T + p^{17} T^{2} \) |
| 79 | \( 1 + 9774477292907752 T + p^{17} T^{2} \) |
| 83 | \( 1 + 17112919183614396 T + p^{17} T^{2} \) |
| 89 | \( 1 + 34698182155846650 T + p^{17} T^{2} \) |
| 97 | \( 1 - 68616916871806082 T + p^{17} T^{2} \) |
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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
−12.44622235206112285559001793724, −10.64474797874451082270736850196, −9.739459960973580396958659648902, −8.803777340303124853090703628380, −6.75830089233701410785373644241, −5.99511973468298344573475307962, −4.57100594659789773628982808208, −2.61924446822476558447326446652, −1.77755707797714418345630236762, 0,
1.77755707797714418345630236762, 2.61924446822476558447326446652, 4.57100594659789773628982808208, 5.99511973468298344573475307962, 6.75830089233701410785373644241, 8.803777340303124853090703628380, 9.739459960973580396958659648902, 10.64474797874451082270736850196, 12.44622235206112285559001793724
