| L(s) = 1 | − 2.44i·5-s − 3.46i·7-s − 2.82·11-s − 3.46·13-s − 1.41i·17-s + 4i·19-s − 4.89·23-s − 0.999·25-s − 2.44i·29-s − 3.46i·31-s − 8.48·35-s + 1.41i·41-s + 8i·43-s − 4.89·47-s − 4.99·49-s + ⋯ |
| L(s) = 1 | − 1.09i·5-s − 1.30i·7-s − 0.852·11-s − 0.960·13-s − 0.342i·17-s + 0.917i·19-s − 1.02·23-s − 0.199·25-s − 0.454i·29-s − 0.622i·31-s − 1.43·35-s + 0.220i·41-s + 1.21i·43-s − 0.714·47-s − 0.714·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.816 - 0.577i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.816 - 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3954160690\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3954160690\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 2.44iT - 5T^{2} \) |
| 7 | \( 1 + 3.46iT - 7T^{2} \) |
| 11 | \( 1 + 2.82T + 11T^{2} \) |
| 13 | \( 1 + 3.46T + 13T^{2} \) |
| 17 | \( 1 + 1.41iT - 17T^{2} \) |
| 19 | \( 1 - 4iT - 19T^{2} \) |
| 23 | \( 1 + 4.89T + 23T^{2} \) |
| 29 | \( 1 + 2.44iT - 29T^{2} \) |
| 31 | \( 1 + 3.46iT - 31T^{2} \) |
| 37 | \( 1 + 37T^{2} \) |
| 41 | \( 1 - 1.41iT - 41T^{2} \) |
| 43 | \( 1 - 8iT - 43T^{2} \) |
| 47 | \( 1 + 4.89T + 47T^{2} \) |
| 53 | \( 1 - 7.34iT - 53T^{2} \) |
| 59 | \( 1 + 11.3T + 59T^{2} \) |
| 61 | \( 1 - 13.8T + 61T^{2} \) |
| 67 | \( 1 - 4iT - 67T^{2} \) |
| 71 | \( 1 - 14.6T + 71T^{2} \) |
| 73 | \( 1 - 4T + 73T^{2} \) |
| 79 | \( 1 - 3.46iT - 79T^{2} \) |
| 83 | \( 1 + 14.1T + 83T^{2} \) |
| 89 | \( 1 + 7.07iT - 89T^{2} \) |
| 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
−8.299992870185084081534505170668, −7.87681036074935821524913651179, −7.20106947403847231512602718402, −6.14138736774396418224118965788, −5.20860164520614199194260377145, −4.53817432341698563169751481332, −3.83270553963726129753682811390, −2.54285694145788918718848048679, −1.28450922634097805119543351516, −0.13314343642974912327358972502,
2.14771643175790448642047251942, 2.63496760325177684961811220094, 3.55593779632051051142966177682, 4.93233902124559041795214033231, 5.47493586367875746970360407252, 6.45509249329875448754298462224, 7.06253818115603818465661952349, 7.946486298181788147008415323627, 8.655256832910570190885127453598, 9.509953379339514726879545710859
