L(s) = 1 | + i·7-s − 1.41i·11-s + i·13-s + 1.41·17-s − 19-s + 1.41·23-s + 1.41i·29-s + 31-s + i·43-s − 1.41·47-s − 1.41i·59-s + 61-s + i·67-s + 1.41·77-s − 1.41·83-s + ⋯ |
L(s) = 1 | + i·7-s − 1.41i·11-s + i·13-s + 1.41·17-s − 19-s + 1.41·23-s + 1.41i·29-s + 31-s + i·43-s − 1.41·47-s − 1.41i·59-s + 61-s + i·67-s + 1.41·77-s − 1.41·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.881 - 0.472i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.881 - 0.472i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.171917278\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.171917278\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( 1 - iT - T^{2} \) |
| 11 | \( 1 + 1.41iT - T^{2} \) |
| 13 | \( 1 - iT - T^{2} \) |
| 17 | \( 1 - 1.41T + T^{2} \) |
| 19 | \( 1 + T + T^{2} \) |
| 23 | \( 1 - 1.41T + T^{2} \) |
| 29 | \( 1 - 1.41iT - T^{2} \) |
| 31 | \( 1 - T + T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - iT - T^{2} \) |
| 47 | \( 1 + 1.41T + T^{2} \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( 1 + 1.41iT - T^{2} \) |
| 61 | \( 1 - T + T^{2} \) |
| 67 | \( 1 - iT - T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 + T^{2} \) |
| 83 | \( 1 + 1.41T + T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + iT - T^{2} \) |
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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
−9.396743263738208099911690800017, −8.625066010674085733002667837817, −8.281688351888695183429605264067, −7.04093209317395606947563179057, −6.28292280200242889734786068729, −5.53844628499141853207671980768, −4.73731543721995202743217354000, −3.46313536411105227985050275299, −2.75623640170741714678183942917, −1.37440935904050908754920087612,
1.05848232644728605482463645322, 2.46052255873522522106509496076, 3.57557658521357573758159374877, 4.46968798262192134968937288485, 5.23004398713219354713466883554, 6.30258738513444534841687974791, 7.18415047070095548953211560849, 7.70941696810620738849954989921, 8.495122324399266807889997268020, 9.652898148737461879963248236734