| L(s) = 1 | + 0.898·5-s − 2.82i·7-s − 4.38i·11-s + 13.7·13-s − 17.5·17-s + 4.38i·19-s + 22.0i·23-s − 24.1·25-s − 44.4·29-s − 53.1i·31-s − 2.54i·35-s − 35.1·37-s + 37.5·41-s − 49.6i·43-s + 38.4i·47-s + ⋯ |
| L(s) = 1 | + 0.179·5-s − 0.404i·7-s − 0.398i·11-s + 1.06·13-s − 1.03·17-s + 0.230i·19-s + 0.958i·23-s − 0.967·25-s − 1.53·29-s − 1.71i·31-s − 0.0726i·35-s − 0.951·37-s + 0.916·41-s − 1.15i·43-s + 0.818i·47-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
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.2514376209\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2514376209\) |
| \(L(2)\) |
|
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 | \( 1 - 0.898T + 25T^{2} \) |
| 7 | \( 1 + 2.82iT - 49T^{2} \) |
| 11 | \( 1 + 4.38iT - 121T^{2} \) |
| 13 | \( 1 - 13.7T + 169T^{2} \) |
| 17 | \( 1 + 17.5T + 289T^{2} \) |
| 19 | \( 1 - 4.38iT - 361T^{2} \) |
| 23 | \( 1 - 22.0iT - 529T^{2} \) |
| 29 | \( 1 + 44.4T + 841T^{2} \) |
| 31 | \( 1 + 53.1iT - 961T^{2} \) |
| 37 | \( 1 + 35.1T + 1.36e3T^{2} \) |
| 41 | \( 1 - 37.5T + 1.68e3T^{2} \) |
| 43 | \( 1 + 49.6iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 38.4iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 1.70T + 2.80e3T^{2} \) |
| 59 | \( 1 + 34.6iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 24.4T + 3.72e3T^{2} \) |
| 67 | \( 1 - 93.7iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 123. iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 10T + 5.32e3T^{2} \) |
| 79 | \( 1 + 131. iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 110. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 73.1T + 7.92e3T^{2} \) |
| 97 | \( 1 + 105.T + 9.40e3T^{2} \) |
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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
−8.459254019482072786334807517818, −7.66363993555265576021055112741, −6.94541825447570038487187501165, −5.92198900208118891268538536386, −5.54434820360929871715459929418, −4.08994213159793036001484222641, −3.76822802458099101720611924561, −2.40876382209212303335308704177, −1.39920039701782216762236992909, −0.05826042087447922638100226593,
1.50282588341295675273000643576, 2.40110886849802700726678536112, 3.52499712893403836850702346764, 4.39630367725870090070468889822, 5.30880967976045009580423914662, 6.14395161922198348454472357513, 6.79838153807363399583405070408, 7.68427066211820741723480765602, 8.649671182330303119683805532850, 9.027711713143379523973364456226
