| L(s) = 1 | − 1.41i·3-s + (1.22 + 1.87i)5-s − 2.64·7-s + 0.999·9-s + 3.46i·11-s + 2.44·13-s + (2.64 − 1.73i)15-s + 4.89·17-s + 6.48·19-s + 3.74i·21-s + (−2 + 4.58i)25-s − 5.65i·27-s − 6·29-s + 4.89·33-s + (−3.24 − 4.94i)35-s + ⋯ |
| L(s) = 1 | − 0.816i·3-s + (0.547 + 0.836i)5-s − 0.999·7-s + 0.333·9-s + 1.04i·11-s + 0.679·13-s + (0.683 − 0.447i)15-s + 1.18·17-s + 1.48·19-s + 0.816i·21-s + (−0.400 + 0.916i)25-s − 1.08i·27-s − 1.11·29-s + 0.852·33-s + (−0.547 − 0.836i)35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0560i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 - 0.0560i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.58816 + 0.0445128i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.58816 + 0.0445128i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (-1.22 - 1.87i)T \) |
| 7 | \( 1 + 2.64T \) |
| good | 3 | \( 1 + 1.41iT - 3T^{2} \) |
| 11 | \( 1 - 3.46iT - 11T^{2} \) |
| 13 | \( 1 - 2.44T + 13T^{2} \) |
| 17 | \( 1 - 4.89T + 17T^{2} \) |
| 19 | \( 1 - 6.48T + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 9.16iT - 37T^{2} \) |
| 41 | \( 1 - 7.48iT - 41T^{2} \) |
| 43 | \( 1 - 5.29T + 43T^{2} \) |
| 47 | \( 1 - 2.82iT - 47T^{2} \) |
| 53 | \( 1 - 9.16iT - 53T^{2} \) |
| 59 | \( 1 - 6.48T + 59T^{2} \) |
| 61 | \( 1 + 11.2iT - 61T^{2} \) |
| 67 | \( 1 + 5.29T + 67T^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 + 9.79T + 73T^{2} \) |
| 79 | \( 1 - 6.92iT - 79T^{2} \) |
| 83 | \( 1 + 9.89iT - 83T^{2} \) |
| 89 | \( 1 + 7.48iT - 89T^{2} \) |
| 97 | \( 1 + 14.6T + 97T^{2} \) |
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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
−10.65526813494867513321731141542, −9.780335950135681541188553979744, −9.363057979458680591291475414714, −7.60379628868593037520190774515, −7.29419488642240578554825912969, −6.31562616381611992208075784230, −5.54960293041176464963708945000, −3.83851641411248022695116881666, −2.75971380147607388699581079945, −1.43404820664341948187188868697,
1.12311863694208111746431873092, 3.17181682155549033725299673872, 3.95029683481274772550798178628, 5.35789777008926265232689228750, 5.82227861232125035640336882043, 7.12899273129523394080231602147, 8.354642719882790980798398986605, 9.240984204332219896256428375478, 9.794268605929240602731181010544, 10.47420264365467112923317219546
