| L(s) = 1 | + 3i·3-s + 7i·7-s − 9·9-s − 44·11-s − 42i·13-s + 94i·17-s + 36·19-s − 21·21-s + 24i·23-s − 27i·27-s − 54·29-s − 112·31-s − 132i·33-s + 322i·37-s + 126·39-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s + 0.377i·7-s − 0.333·9-s − 1.20·11-s − 0.896i·13-s + 1.34i·17-s + 0.434·19-s − 0.218·21-s + 0.217i·23-s − 0.192i·27-s − 0.345·29-s − 0.648·31-s − 0.696i·33-s + 1.43i·37-s + 0.517·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.7897121771\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7897121771\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 3iT \) |
| 5 | \( 1 \) |
| 7 | \( 1 - 7iT \) |
| good | 11 | \( 1 + 44T + 1.33e3T^{2} \) |
| 13 | \( 1 + 42iT - 2.19e3T^{2} \) |
| 17 | \( 1 - 94iT - 4.91e3T^{2} \) |
| 19 | \( 1 - 36T + 6.85e3T^{2} \) |
| 23 | \( 1 - 24iT - 1.21e4T^{2} \) |
| 29 | \( 1 + 54T + 2.43e4T^{2} \) |
| 31 | \( 1 + 112T + 2.97e4T^{2} \) |
| 37 | \( 1 - 322iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 22T + 6.89e4T^{2} \) |
| 43 | \( 1 - 292iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 272iT - 1.03e5T^{2} \) |
| 53 | \( 1 + 578iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 44T + 2.05e5T^{2} \) |
| 61 | \( 1 + 26T + 2.26e5T^{2} \) |
| 67 | \( 1 + 12iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 280T + 3.57e5T^{2} \) |
| 73 | \( 1 - 410iT - 3.89e5T^{2} \) |
| 79 | \( 1 - 320T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.25e3iT - 5.71e5T^{2} \) |
| 89 | \( 1 - 38T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.25e3iT - 9.12e5T^{2} \) |
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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
−8.367797657500315089446934628883, −8.160444190253921860335272609636, −7.12498584967235574607473051650, −5.98477967381715108111743603738, −5.44351844288534938206767737478, −4.67710112914054160229110391454, −3.54684309730237900498135863804, −2.86344895080733436927374339164, −1.70778359793285950506790058799, −0.19492181758755992352065068663,
0.815913654587610623421568016160, 2.07033761478680589607436521014, 2.85652106367714484015391532714, 3.98413895943466321878384248693, 5.00236208487420171675133693192, 5.66881453358045921397345767967, 6.72865505184725956209346574752, 7.40701385178449276578977118544, 7.84165732602193684236429823508, 8.988536803593879439551658037825
