| L(s) = 1 | + 4.24i·5-s + 8.48·7-s − 4i·11-s + 18·13-s + 4.24i·17-s − 16.9·19-s + 36i·23-s + 7.00·25-s − 12.7i·29-s + 8.48·31-s + 35.9i·35-s + 36·37-s + 29.6i·41-s + 67.8·43-s + 36i·47-s + ⋯ |
| L(s) = 1 | + 0.848i·5-s + 1.21·7-s − 0.363i·11-s + 1.38·13-s + 0.249i·17-s − 0.893·19-s + 1.56i·23-s + 0.280·25-s − 0.438i·29-s + 0.273·31-s + 1.02i·35-s + 0.972·37-s + 0.724i·41-s + 1.57·43-s + 0.765i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.659060214\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.659060214\) |
| \(L(2)\) |
|
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 | \( 1 - 4.24iT - 25T^{2} \) |
| 7 | \( 1 - 8.48T + 49T^{2} \) |
| 11 | \( 1 + 4iT - 121T^{2} \) |
| 13 | \( 1 - 18T + 169T^{2} \) |
| 17 | \( 1 - 4.24iT - 289T^{2} \) |
| 19 | \( 1 + 16.9T + 361T^{2} \) |
| 23 | \( 1 - 36iT - 529T^{2} \) |
| 29 | \( 1 + 12.7iT - 841T^{2} \) |
| 31 | \( 1 - 8.48T + 961T^{2} \) |
| 37 | \( 1 - 36T + 1.36e3T^{2} \) |
| 41 | \( 1 - 29.6iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 67.8T + 1.84e3T^{2} \) |
| 47 | \( 1 - 36iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 80.6iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 80iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 36T + 3.72e3T^{2} \) |
| 67 | \( 1 + 118.T + 4.48e3T^{2} \) |
| 71 | \( 1 + 108iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 56T + 5.32e3T^{2} \) |
| 79 | \( 1 - 25.4T + 6.24e3T^{2} \) |
| 83 | \( 1 - 76iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 89.0iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 104T + 9.40e3T^{2} \) |
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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
−8.837288320493050897118544766628, −8.055054105105901799482705432429, −7.61192175233753659341059369730, −6.44484920642102139890187483322, −6.00199216645760551865128171124, −4.95273144512609675010603710781, −4.02368623284356445703828905157, −3.21265809862949163570068451340, −2.06695141705515783611716983226, −1.09303320892179352384823201806,
0.76741683850571098636273368182, 1.62177220926973201313887640709, 2.71787949717832760495332015342, 4.30114877837366467979853018582, 4.43498183139070041143050492129, 5.52510147718540070394885442421, 6.26291245538355357639567832566, 7.28208857485663607778813446440, 8.134473256308346194189262805476, 8.735048960037929886471056104793
