| L(s) = 1 | + (−1.70 − 0.323i)3-s + 1.55i·5-s + (2.79 + 1.09i)9-s + 2.53·13-s + (0.502 − 2.64i)15-s + 4.33i·17-s + 6.83i·19-s − 3.80·23-s + 2.58·25-s + (−4.39 − 2.77i)27-s − 8.33i·29-s − 4.89i·31-s + 1.58·37-s + (−4.31 − 0.818i)39-s + 7.44i·41-s + ⋯ |
| L(s) = 1 | + (−0.982 − 0.186i)3-s + 0.695i·5-s + (0.930 + 0.366i)9-s + 0.702·13-s + (0.129 − 0.683i)15-s + 1.05i·17-s + 1.56i·19-s − 0.794·23-s + 0.516·25-s + (−0.845 − 0.533i)27-s − 1.54i·29-s − 0.879i·31-s + 0.260·37-s + (−0.690 − 0.131i)39-s + 1.16i·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.329 - 0.944i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.329 - 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9634290630\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9634290630\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.70 + 0.323i)T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 1.55iT - 5T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 2.53T + 13T^{2} \) |
| 17 | \( 1 - 4.33iT - 17T^{2} \) |
| 19 | \( 1 - 6.83iT - 19T^{2} \) |
| 23 | \( 1 + 3.80T + 23T^{2} \) |
| 29 | \( 1 + 8.33iT - 29T^{2} \) |
| 31 | \( 1 + 4.89iT - 31T^{2} \) |
| 37 | \( 1 - 1.58T + 37T^{2} \) |
| 41 | \( 1 - 7.44iT - 41T^{2} \) |
| 43 | \( 1 + 6.20iT - 43T^{2} \) |
| 47 | \( 1 - 5.38T + 47T^{2} \) |
| 53 | \( 1 - 10.5iT - 53T^{2} \) |
| 59 | \( 1 + 10.2T + 59T^{2} \) |
| 61 | \( 1 + 8.19T + 61T^{2} \) |
| 67 | \( 1 - 3.46iT - 67T^{2} \) |
| 71 | \( 1 - 14.4T + 71T^{2} \) |
| 73 | \( 1 - 11.5T + 73T^{2} \) |
| 79 | \( 1 - 6.92iT - 79T^{2} \) |
| 83 | \( 1 + 4.82T + 83T^{2} \) |
| 89 | \( 1 - 4.33iT - 89T^{2} \) |
| 97 | \( 1 + 9.89T + 97T^{2} \) |
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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
−9.422995215452806275120011598179, −8.083307220679887463414182256597, −7.80623857857204463403412399730, −6.63115373271758509059343407035, −6.11055360472435817905321049112, −5.61724194809316227325828466073, −4.28540018542469691784965079655, −3.74585175542072722046361198366, −2.33131988958023012541209995967, −1.24321048596786767867957207934,
0.42421961734103385154340242545, 1.47157019654962619058528487150, 2.98300108660803064722114112371, 4.11109752176415412283248723403, 5.00153078949759572348679606313, 5.32281639548588826334126583538, 6.50893485260412920927300887955, 6.96208068105549462677667867953, 7.972406034965537610737194577345, 9.072251949445506239629457776359
