| L(s) = 1 | + i·2-s + 3-s − 4-s − i·5-s + i·6-s − i·7-s − i·8-s + 9-s + 10-s + 3i·11-s − 12-s + (3 − 2i)13-s + 14-s − i·15-s + 16-s + 7·17-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s + 0.577·3-s − 0.5·4-s − 0.447i·5-s + 0.408i·6-s − 0.377i·7-s − 0.353i·8-s + 0.333·9-s + 0.316·10-s + 0.904i·11-s − 0.288·12-s + (0.832 − 0.554i)13-s + 0.267·14-s − 0.258i·15-s + 0.250·16-s + 1.69·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.832 - 0.554i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.832 - 0.554i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.70292 + 0.515605i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.70292 + 0.515605i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - iT \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 + iT \) |
| 13 | \( 1 + (-3 + 2i)T \) |
| good | 5 | \( 1 + iT - 5T^{2} \) |
| 11 | \( 1 - 3iT - 11T^{2} \) |
| 17 | \( 1 - 7T + 17T^{2} \) |
| 19 | \( 1 - 3iT - 19T^{2} \) |
| 23 | \( 1 - T + 23T^{2} \) |
| 29 | \( 1 + T + 29T^{2} \) |
| 31 | \( 1 + 8iT - 31T^{2} \) |
| 37 | \( 1 + iT - 37T^{2} \) |
| 41 | \( 1 - 4iT - 41T^{2} \) |
| 43 | \( 1 + 5T + 43T^{2} \) |
| 47 | \( 1 - 47T^{2} \) |
| 53 | \( 1 + 6T + 53T^{2} \) |
| 59 | \( 1 - 10iT - 59T^{2} \) |
| 61 | \( 1 + 13T + 61T^{2} \) |
| 67 | \( 1 + 8iT - 67T^{2} \) |
| 71 | \( 1 - 6iT - 71T^{2} \) |
| 73 | \( 1 + 13iT - 73T^{2} \) |
| 79 | \( 1 + 12T + 79T^{2} \) |
| 83 | \( 1 - 2iT - 83T^{2} \) |
| 89 | \( 1 - 12iT - 89T^{2} \) |
| 97 | \( 1 - 6iT - 97T^{2} \) |
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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
−10.57712839766422400962204337484, −9.833323138282610486409897849216, −9.037207096057459242685360680499, −7.949835094309691188515700603833, −7.61178950661489259753692982695, −6.35093675030024605547797105182, −5.34465390238234539769941603269, −4.27516766227616299875620750773, −3.23123754426432929052188179294, −1.31855408242780957291214467754,
1.38390742784676669259892210015, 2.95746342164692720974343031467, 3.50561592638930415211628589898, 4.94946932056961628938788272653, 6.08086898664347684779911898670, 7.21011486025012968461358392795, 8.393621315860024309714270669955, 8.912382805672646779752354853771, 9.909433120286834848958008835898, 10.76478819585551596315768989104
