| L(s) = 1 | − 4·5-s − 11.3i·7-s + 14.1i·11-s + 20·13-s + 10·17-s + 14.1i·19-s − 11.3i·23-s − 9·25-s − 20·29-s + 45.2i·35-s + 20·37-s − 30·41-s − 2.82i·43-s + 67.8i·47-s − 79.0·49-s + ⋯ |
| L(s) = 1 | − 0.800·5-s − 1.61i·7-s + 1.28i·11-s + 1.53·13-s + 0.588·17-s + 0.744i·19-s − 0.491i·23-s − 0.359·25-s − 0.689·29-s + 1.29i·35-s + 0.540·37-s − 0.731·41-s − 0.0657i·43-s + 1.44i·47-s − 1.61·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.732701915\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.732701915\) |
| \(L(2)\) |
|
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 | \( 1 + 4T + 25T^{2} \) |
| 7 | \( 1 + 11.3iT - 49T^{2} \) |
| 11 | \( 1 - 14.1iT - 121T^{2} \) |
| 13 | \( 1 - 20T + 169T^{2} \) |
| 17 | \( 1 - 10T + 289T^{2} \) |
| 19 | \( 1 - 14.1iT - 361T^{2} \) |
| 23 | \( 1 + 11.3iT - 529T^{2} \) |
| 29 | \( 1 + 20T + 841T^{2} \) |
| 31 | \( 1 - 961T^{2} \) |
| 37 | \( 1 - 20T + 1.36e3T^{2} \) |
| 41 | \( 1 + 30T + 1.68e3T^{2} \) |
| 43 | \( 1 + 2.82iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 67.8iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 60T + 2.80e3T^{2} \) |
| 59 | \( 1 + 42.4iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 28T + 3.72e3T^{2} \) |
| 67 | \( 1 - 82.0iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 56.5iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 10T + 5.32e3T^{2} \) |
| 79 | \( 1 - 113. iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 25.4iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 22T + 7.92e3T^{2} \) |
| 97 | \( 1 - 150T + 9.40e3T^{2} \) |
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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
−8.667599103170719997838009395825, −7.86381253846379394250593175967, −7.41852044335674216715256851285, −6.68112255914800431692252777346, −5.74513256137580035355323991968, −4.47979563917095382918232565682, −4.01757048219578845455249827276, −3.34904623498636734048225918846, −1.71901595482786563279102616159, −0.76767039936684423367409329050,
0.63448632540110217153682705338, 1.99948542631572431683018170011, 3.24021065722272968572757434491, 3.64892229327472694461089165781, 4.97871121563969503683805422051, 5.84896518757649837806522123420, 6.18820899048238082691885470898, 7.44051169026616000764895701148, 8.261229264745369855686063292908, 8.751828618565635932371323938952
