| L(s) = 1 | − 2.64i·3-s + 0.508·5-s − i·7-s − 3.97·9-s + 1.38i·11-s − 2.62i·13-s − 1.34i·15-s − 3.63i·17-s − 0.0724i·19-s − 2.64·21-s − 3.11·23-s − 4.74·25-s + 2.57i·27-s − 5.90i·29-s − 2.48·31-s + ⋯ |
| L(s) = 1 | − 1.52i·3-s + 0.227·5-s − 0.377i·7-s − 1.32·9-s + 0.416i·11-s − 0.729i·13-s − 0.346i·15-s − 0.882i·17-s − 0.0166i·19-s − 0.576·21-s − 0.649·23-s − 0.948·25-s + 0.495i·27-s − 1.09i·29-s − 0.446·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.992 + 0.125i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.992 + 0.125i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.217525886\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.217525886\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + iT \) |
| 41 | \( 1 + (0.803 + 6.35i)T \) |
| good | 3 | \( 1 + 2.64iT - 3T^{2} \) |
| 5 | \( 1 - 0.508T + 5T^{2} \) |
| 11 | \( 1 - 1.38iT - 11T^{2} \) |
| 13 | \( 1 + 2.62iT - 13T^{2} \) |
| 17 | \( 1 + 3.63iT - 17T^{2} \) |
| 19 | \( 1 + 0.0724iT - 19T^{2} \) |
| 23 | \( 1 + 3.11T + 23T^{2} \) |
| 29 | \( 1 + 5.90iT - 29T^{2} \) |
| 31 | \( 1 + 2.48T + 31T^{2} \) |
| 37 | \( 1 + 0.477T + 37T^{2} \) |
| 43 | \( 1 + 0.169T + 43T^{2} \) |
| 47 | \( 1 - 7.87iT - 47T^{2} \) |
| 53 | \( 1 - 1.97iT - 53T^{2} \) |
| 59 | \( 1 + 4.22T + 59T^{2} \) |
| 61 | \( 1 - 10.4T + 61T^{2} \) |
| 67 | \( 1 - 8.77iT - 67T^{2} \) |
| 71 | \( 1 + 0.853iT - 71T^{2} \) |
| 73 | \( 1 - 6.17T + 73T^{2} \) |
| 79 | \( 1 - 1.30iT - 79T^{2} \) |
| 83 | \( 1 - 5.00T + 83T^{2} \) |
| 89 | \( 1 + 6.89iT - 89T^{2} \) |
| 97 | \( 1 - 9.36iT - 97T^{2} \) |
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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
−9.425461246532561870464563557563, −8.262741266252572499184766185532, −7.65146598369117010878052840332, −7.05417586290315719519522798775, −6.16565927994196765093214300683, −5.40449597617378375614972340008, −4.08723563425972592904064418456, −2.72065361957388554484780067297, −1.80143964719722051418365463241, −0.51071593180046766742210540486,
1.95437674838664645962183510697, 3.37054791047465530830926139253, 4.04012169505857968513069891237, 5.02081785490427582281449205637, 5.77079583710536698698656353455, 6.66693757131656621516645114292, 8.028783240129131753738276795245, 8.790271093377774657886739019398, 9.466085091107214018634292392098, 10.10796894026382923921262684382
