| L(s) = 1 | + (1.58 + 0.707i)3-s − 4.24i·7-s + (2.00 + 2.23i)9-s + (3 − 6.70i)21-s + 9.48·23-s + (1.58 + 4.94i)27-s − 8.94i·29-s + 4.47i·41-s − 12.7i·43-s + 9.48·47-s − 10.9·49-s − 8·61-s + (9.48 − 8.48i)63-s + 4.24i·67-s + (15.0 + 6.70i)69-s + ⋯ |
| L(s) = 1 | + (0.912 + 0.408i)3-s − 1.60i·7-s + (0.666 + 0.745i)9-s + (0.654 − 1.46i)21-s + 1.97·23-s + (0.304 + 0.952i)27-s − 1.66i·29-s + 0.698i·41-s − 1.94i·43-s + 1.38·47-s − 1.57·49-s − 1.02·61-s + (1.19 − 1.06i)63-s + 0.518i·67-s + (1.80 + 0.807i)69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.912 + 0.408i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.912 + 0.408i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.331005848\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.331005848\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.58 - 0.707i)T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 4.24iT - 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 23 | \( 1 - 9.48T + 23T^{2} \) |
| 29 | \( 1 + 8.94iT - 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 + 37T^{2} \) |
| 41 | \( 1 - 4.47iT - 41T^{2} \) |
| 43 | \( 1 + 12.7iT - 43T^{2} \) |
| 47 | \( 1 - 9.48T + 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 8T + 61T^{2} \) |
| 67 | \( 1 - 4.24iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 73T^{2} \) |
| 79 | \( 1 - 79T^{2} \) |
| 83 | \( 1 - 9.48T + 83T^{2} \) |
| 89 | \( 1 - 17.8iT - 89T^{2} \) |
| 97 | \( 1 + 97T^{2} \) |
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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
−9.642591262140849168317303512474, −8.978087256848642350971043431372, −7.978169787560015087918798535045, −7.36787595225732594135510627652, −6.65435438878396866518088488243, −5.19478310939508148058502465999, −4.27312406216852418573863544590, −3.63487922013968776080021748011, −2.52409864054875588747354988307, −1.02500644120511307670571005275,
1.46242598362116845334824100896, 2.64654296774814580417924265032, 3.24408181734925928681211336652, 4.69008387541025363264653674050, 5.60907769289998755106605175634, 6.58856874111609122198008897354, 7.39194305892170427854402988491, 8.348986782664955408265497712229, 9.073521363571290174084166347157, 9.301250554805345830268443050309
