| L(s) = 1 | + 3-s + (1.22 + 1.87i)5-s − 2·9-s − 4.89·11-s − 4.58i·13-s + (1.22 + 1.87i)15-s − 2.44·17-s − 2.44·19-s + (−3 + 3.74i)23-s + (−2 + 4.58i)25-s − 5·27-s − 3·29-s + 4.58i·31-s − 4.89·33-s + 4.89·37-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + (0.547 + 0.836i)5-s − 0.666·9-s − 1.47·11-s − 1.27i·13-s + (0.316 + 0.483i)15-s − 0.594·17-s − 0.561·19-s + (−0.625 + 0.780i)23-s + (−0.400 + 0.916i)25-s − 0.962·27-s − 0.557·29-s + 0.823i·31-s − 0.852·33-s + 0.805·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 - 0.0960i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.995 - 0.0960i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4042211149\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4042211149\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (-1.22 - 1.87i)T \) |
| 23 | \( 1 + (3 - 3.74i)T \) |
| good | 3 | \( 1 - T + 3T^{2} \) |
| 7 | \( 1 - 7T^{2} \) |
| 11 | \( 1 + 4.89T + 11T^{2} \) |
| 13 | \( 1 + 4.58iT - 13T^{2} \) |
| 17 | \( 1 + 2.44T + 17T^{2} \) |
| 19 | \( 1 + 2.44T + 19T^{2} \) |
| 29 | \( 1 + 3T + 29T^{2} \) |
| 31 | \( 1 - 4.58iT - 31T^{2} \) |
| 37 | \( 1 - 4.89T + 37T^{2} \) |
| 41 | \( 1 + 3T + 41T^{2} \) |
| 43 | \( 1 - 11.2iT - 43T^{2} \) |
| 47 | \( 1 + 9T + 47T^{2} \) |
| 53 | \( 1 + 4.89T + 53T^{2} \) |
| 59 | \( 1 + 9.16iT - 59T^{2} \) |
| 61 | \( 1 - 11.2iT - 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 + 13.7iT - 71T^{2} \) |
| 73 | \( 1 + 4.58iT - 73T^{2} \) |
| 79 | \( 1 + 7.34T + 79T^{2} \) |
| 83 | \( 1 + 3.74iT - 83T^{2} \) |
| 89 | \( 1 + 3.74iT - 89T^{2} \) |
| 97 | \( 1 - 9.79T + 97T^{2} \) |
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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
−9.710435337968110491174930506724, −8.789598801437329937897341846002, −7.920384720271692897419256775350, −7.57422900144387181536724320038, −6.32191362593348365794404325148, −5.71120169066776123779638387291, −4.87528081723599115190879580249, −3.38263240706867473309418610355, −2.85686275816703161421351086564, −2.00231717390234486346829632302,
0.11866920241489012450222362196, 2.02074777555634067416883342830, 2.50161683764124524329507681317, 3.92790280088325373947609346548, 4.75502408066513450002033873557, 5.61847347114298276627142261224, 6.36593492147629343623786684738, 7.46547264232925534822528320905, 8.369471043430888707803548604073, 8.715694221292644285687249893231
