| L(s) = 1 | − 2.44·5-s + 2.44·7-s + (−3 − 1.41i)11-s + 4.24i·13-s − 3.46i·17-s + 1.41i·23-s + 0.999·25-s − 6.92i·29-s − 3.46i·31-s − 5.99·35-s − 2·37-s − 6.92i·41-s − 9.79·43-s + 7.07i·47-s − 1.00·49-s + ⋯ |
| L(s) = 1 | − 1.09·5-s + 0.925·7-s + (−0.904 − 0.426i)11-s + 1.17i·13-s − 0.840i·17-s + 0.294i·23-s + 0.199·25-s − 1.28i·29-s − 0.622i·31-s − 1.01·35-s − 0.328·37-s − 1.08i·41-s − 1.49·43-s + 1.03i·47-s − 0.142·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1584 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.821 + 0.570i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1584 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.821 + 0.570i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4108520420\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4108520420\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 11 | \( 1 + (3 + 1.41i)T \) |
| good | 5 | \( 1 + 2.44T + 5T^{2} \) |
| 7 | \( 1 - 2.44T + 7T^{2} \) |
| 13 | \( 1 - 4.24iT - 13T^{2} \) |
| 17 | \( 1 + 3.46iT - 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 - 1.41iT - 23T^{2} \) |
| 29 | \( 1 + 6.92iT - 29T^{2} \) |
| 31 | \( 1 + 3.46iT - 31T^{2} \) |
| 37 | \( 1 + 2T + 37T^{2} \) |
| 41 | \( 1 + 6.92iT - 41T^{2} \) |
| 43 | \( 1 + 9.79T + 43T^{2} \) |
| 47 | \( 1 - 7.07iT - 47T^{2} \) |
| 53 | \( 1 + 7.34T + 53T^{2} \) |
| 59 | \( 1 - 14.1iT - 59T^{2} \) |
| 61 | \( 1 + 12.7iT - 61T^{2} \) |
| 67 | \( 1 - 3.46iT - 67T^{2} \) |
| 71 | \( 1 + 9.89iT - 71T^{2} \) |
| 73 | \( 1 + 8.48iT - 73T^{2} \) |
| 79 | \( 1 + 2.44T + 79T^{2} \) |
| 83 | \( 1 + 12T + 83T^{2} \) |
| 89 | \( 1 + 4.89T + 89T^{2} \) |
| 97 | \( 1 + 8T + 97T^{2} \) |
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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
−8.989636506699144145994908240530, −8.091503881028972151712367079604, −7.71535090813829932447857761569, −6.86636027305826951601568235003, −5.73354480490664087015383566364, −4.75954510626149368388795380823, −4.16453011900442013822460644010, −3.04222734005125359646225355926, −1.82742892418122828755979362369, −0.15912832029506996456657498644,
1.50399431092693880616321279907, 2.90373584091771913066129078834, 3.81343859589674030550127513939, 4.86906495872609416435022107462, 5.37073511413163294598744345430, 6.68259322183065217844392105458, 7.56275378121798654793317196938, 8.208324609181004525691011806588, 8.505945726234293845764137018406, 9.934667170130887122742962934324
