L(s) = 1 | + 1.41i·2-s + 1.41i·3-s − 1.00·4-s + 5-s − 2.00·6-s + 7-s − 1.00·9-s + 1.41i·10-s − 11-s − 1.41i·12-s − 1.41i·13-s + 1.41i·14-s + 1.41i·15-s − 0.999·16-s − 1.41i·17-s − 1.41i·18-s + ⋯ |
L(s) = 1 | + 1.41i·2-s + 1.41i·3-s − 1.00·4-s + 5-s − 2.00·6-s + 7-s − 1.00·9-s + 1.41i·10-s − 11-s − 1.41i·12-s − 1.41i·13-s + 1.41i·14-s + 1.41i·15-s − 0.999·16-s − 1.41i·17-s − 1.41i·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1099 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1099 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.255061004\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.255061004\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 - T \) |
| 157 | \( 1 + T \) |
good | 2 | \( 1 - 1.41iT - T^{2} \) |
| 3 | \( 1 - 1.41iT - T^{2} \) |
| 5 | \( 1 - T + T^{2} \) |
| 11 | \( 1 + T + T^{2} \) |
| 13 | \( 1 + 1.41iT - T^{2} \) |
| 17 | \( 1 + 1.41iT - T^{2} \) |
| 19 | \( 1 - 1.41iT - T^{2} \) |
| 23 | \( 1 - 1.41iT - T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 + 1.41iT - T^{2} \) |
| 37 | \( 1 + T + T^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 + 1.41iT - T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 - T + T^{2} \) |
| 61 | \( 1 - T + T^{2} \) |
| 67 | \( 1 - T + T^{2} \) |
| 71 | \( 1 + T + T^{2} \) |
| 73 | \( 1 - T + T^{2} \) |
| 79 | \( 1 + 1.41iT - T^{2} \) |
| 83 | \( 1 + T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + T + T^{2} \) |
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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
−10.14708363041210456982881773240, −9.684516409772640015859514790217, −8.719103325454428039134100272174, −7.904838050419210420809007195296, −7.34144124055477092027925249526, −5.77791151515662225600748266041, −5.41150114349308182059975919530, −5.00137227226463542801986455885, −3.69409142897207908743715002290, −2.28873323938685465760522650434,
1.32775542764724275912616004579, 2.04019478288948419773736791401, 2.60809052647635850507664761521, 4.25961002387416376719613084651, 5.22311812951748254059338341638, 6.50157303852040334319083637926, 6.96812051134157432931337368838, 8.251207927507114370158544227628, 8.808563190495100538701679310691, 9.888377561001886859564733883998