| L(s) = 1 | + i·3-s − 0.0802·7-s − 9-s − 2.41i·11-s − 5.26i·13-s − 0.255·17-s + 6.95i·19-s − 0.0802i·21-s + 1.64·23-s − i·27-s − 4.51i·29-s − 8.29·31-s + 2.41·33-s − 2.67i·37-s + 5.26·39-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s − 0.0303·7-s − 0.333·9-s − 0.728i·11-s − 1.46i·13-s − 0.0620·17-s + 1.59i·19-s − 0.0175i·21-s + 0.343·23-s − 0.192i·27-s − 0.838i·29-s − 1.48·31-s + 0.420·33-s − 0.439i·37-s + 0.843·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.136 + 0.990i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.136 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9159213115\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9159213115\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 0.0802T + 7T^{2} \) |
| 11 | \( 1 + 2.41iT - 11T^{2} \) |
| 13 | \( 1 + 5.26iT - 13T^{2} \) |
| 17 | \( 1 + 0.255T + 17T^{2} \) |
| 19 | \( 1 - 6.95iT - 19T^{2} \) |
| 23 | \( 1 - 1.64T + 23T^{2} \) |
| 29 | \( 1 + 4.51iT - 29T^{2} \) |
| 31 | \( 1 + 8.29T + 31T^{2} \) |
| 37 | \( 1 + 2.67iT - 37T^{2} \) |
| 41 | \( 1 + 8.11T + 41T^{2} \) |
| 43 | \( 1 + 4.08iT - 43T^{2} \) |
| 47 | \( 1 + 5.70T + 47T^{2} \) |
| 53 | \( 1 + 11.5iT - 53T^{2} \) |
| 59 | \( 1 - 12.6iT - 59T^{2} \) |
| 61 | \( 1 + 11.9iT - 61T^{2} \) |
| 67 | \( 1 + 7.27iT - 67T^{2} \) |
| 71 | \( 1 - 11.3T + 71T^{2} \) |
| 73 | \( 1 + 12.0T + 73T^{2} \) |
| 79 | \( 1 + 5.50T + 79T^{2} \) |
| 83 | \( 1 + 9.20iT - 83T^{2} \) |
| 89 | \( 1 - 11.9T + 89T^{2} \) |
| 97 | \( 1 - 8.50T + 97T^{2} \) |
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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
−8.625519098941604999322154058169, −8.148134344358782961975012954770, −7.36336987147359297836058709394, −6.17748331592262102099720097855, −5.61993358563640664339101201761, −4.88277593019421229833474446764, −3.61103367400549340136668523370, −3.28378868875170002378318391871, −1.86854674188397311144412374046, −0.30240438525546169422506466893,
1.41185093504927927469427605674, 2.28980633632573367596766472322, 3.36920201798437275889921775531, 4.53317987221529898463913602580, 5.10680778579785900957803175296, 6.31891591704467038787767878215, 6.96338665309887552129017672563, 7.36511818445749108205384892505, 8.510135132818261655952975345330, 9.124046607310043922110411536186
