| L(s) = 1 | − i·3-s + (−1 + 2i)5-s + i·7-s − 9-s + 11-s − 4i·13-s + (2 + i)15-s − i·17-s − 19-s + 21-s + 4i·23-s + (−3 − 4i)25-s + i·27-s + 29-s − 6·31-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s + (−0.447 + 0.894i)5-s + 0.377i·7-s − 0.333·9-s + 0.301·11-s − 1.10i·13-s + (0.516 + 0.258i)15-s − 0.242i·17-s − 0.229·19-s + 0.218·21-s + 0.834i·23-s + (−0.600 − 0.800i)25-s + 0.192i·27-s + 0.185·29-s − 1.07·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8348912501\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8348912501\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (1 - 2i)T \) |
| 17 | \( 1 + iT \) |
| good | 7 | \( 1 - iT - 7T^{2} \) |
| 11 | \( 1 - T + 11T^{2} \) |
| 13 | \( 1 + 4iT - 13T^{2} \) |
| 19 | \( 1 + T + 19T^{2} \) |
| 23 | \( 1 - 4iT - 23T^{2} \) |
| 29 | \( 1 - T + 29T^{2} \) |
| 31 | \( 1 + 6T + 31T^{2} \) |
| 37 | \( 1 + 5iT - 37T^{2} \) |
| 41 | \( 1 + T + 41T^{2} \) |
| 43 | \( 1 - 6iT - 43T^{2} \) |
| 47 | \( 1 + 3iT - 47T^{2} \) |
| 53 | \( 1 - iT - 53T^{2} \) |
| 59 | \( 1 - 6T + 59T^{2} \) |
| 61 | \( 1 + 4T + 61T^{2} \) |
| 67 | \( 1 + 2iT - 67T^{2} \) |
| 71 | \( 1 + 8T + 71T^{2} \) |
| 73 | \( 1 + 5iT - 73T^{2} \) |
| 79 | \( 1 + 14T + 79T^{2} \) |
| 83 | \( 1 + 4iT - 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 + 14iT - 97T^{2} \) |
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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
−8.041889703031213509532581770375, −7.40839818489721870406146526039, −6.90788006775617250412163675601, −5.94035277803158704732672436615, −5.50225489991146238744736326909, −4.28394964200806525511251916053, −3.34921533049855993304681134303, −2.73549029315873466119514107941, −1.68868724762115940085648116846, −0.25730828131478725156877622124,
1.12051980684321038727843655736, 2.24216654335027260203167518383, 3.57565253022825216396980564362, 4.16590019197400274598073153437, 4.73702620550763400793832826214, 5.56867605177513961199848770194, 6.49148928084948667833830848596, 7.22615727048779151424952284749, 8.074141981506668986753095870787, 8.878947011534959073843108946710
