| L(s) = 1 | + 2.23·3-s + (1.58 − 1.58i)5-s + 2.82i·7-s + 2.00·9-s − 5i·13-s + (3.53 − 3.53i)15-s − 3.16·17-s + 7.07·19-s + 6.32i·21-s + (2.23 + 4.24i)23-s − 5.00i·25-s − 2.23·27-s + 9·29-s − 6.70i·31-s + (4.47 + 4.47i)35-s + ⋯ |
| L(s) = 1 | + 1.29·3-s + (0.707 − 0.707i)5-s + 1.06i·7-s + 0.666·9-s − 1.38i·13-s + (0.912 − 0.912i)15-s − 0.766·17-s + 1.62·19-s + 1.38i·21-s + (0.466 + 0.884i)23-s − 1.00i·25-s − 0.430·27-s + 1.67·29-s − 1.20i·31-s + (0.755 + 0.755i)35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.955 + 0.295i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.955 + 0.295i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.180163753\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.180163753\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (-1.58 + 1.58i)T \) |
| 23 | \( 1 + (-2.23 - 4.24i)T \) |
| good | 3 | \( 1 - 2.23T + 3T^{2} \) |
| 7 | \( 1 - 2.82iT - 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + 5iT - 13T^{2} \) |
| 17 | \( 1 + 3.16T + 17T^{2} \) |
| 19 | \( 1 - 7.07T + 19T^{2} \) |
| 29 | \( 1 - 9T + 29T^{2} \) |
| 31 | \( 1 + 6.70iT - 31T^{2} \) |
| 37 | \( 1 + 37T^{2} \) |
| 41 | \( 1 + 3T + 41T^{2} \) |
| 43 | \( 1 + 1.41iT - 43T^{2} \) |
| 47 | \( 1 - 6.70T + 47T^{2} \) |
| 53 | \( 1 - 6.32T + 53T^{2} \) |
| 59 | \( 1 - 4.47iT - 59T^{2} \) |
| 61 | \( 1 - 9.48iT - 61T^{2} \) |
| 67 | \( 1 - 2.82iT - 67T^{2} \) |
| 71 | \( 1 + 6.70iT - 71T^{2} \) |
| 73 | \( 1 - 15iT - 73T^{2} \) |
| 79 | \( 1 + 7.07T + 79T^{2} \) |
| 83 | \( 1 - 12.7iT - 83T^{2} \) |
| 89 | \( 1 + 3.16iT - 89T^{2} \) |
| 97 | \( 1 + 18.9T + 97T^{2} \) |
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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
−9.170810668914946699718052570247, −8.478473670831539483277084068667, −7.995457988763873815549600999454, −7.00219506059498931013246559833, −5.57367851238791287763728457247, −5.49956759344230043996307532420, −4.14008030342416432362190349078, −2.87244567081197519444653261712, −2.53521732429237426541804023422, −1.18070876617604462435200750968,
1.37801035650357699717703156088, 2.47545840151407394914731372555, 3.21834619846016720560897581680, 4.12534146551153829745646198434, 5.06512248379527647516020518955, 6.49669574891062442666683959711, 6.93561624616728864590160870333, 7.65283522088249167578119071580, 8.680608272081509080646319247503, 9.200469141276053138381593900624
