| L(s) = 1 | + (5 + 5i)13-s + (−5 + 5i)17-s − 4i·29-s + (−5 + 5i)37-s − 8·41-s − 7i·49-s + (5 + 5i)53-s − 12·61-s + (−5 − 5i)73-s + 16i·89-s + (−5 + 5i)97-s − 2·101-s + 6i·109-s + (15 + 15i)113-s + ⋯ |
| L(s) = 1 | + (1.38 + 1.38i)13-s + (−1.21 + 1.21i)17-s − 0.742i·29-s + (−0.821 + 0.821i)37-s − 1.24·41-s − i·49-s + (0.686 + 0.686i)53-s − 1.53·61-s + (−0.585 − 0.585i)73-s + 1.69i·89-s + (−0.507 + 0.507i)97-s − 0.199·101-s + 0.574i·109-s + (1.41 + 1.41i)113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.525 - 0.850i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.525 - 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.170800337\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.170800337\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 7iT^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 + (-5 - 5i)T + 13iT^{2} \) |
| 17 | \( 1 + (5 - 5i)T - 17iT^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 - 23iT^{2} \) |
| 29 | \( 1 + 4iT - 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 + (5 - 5i)T - 37iT^{2} \) |
| 41 | \( 1 + 8T + 41T^{2} \) |
| 43 | \( 1 - 43iT^{2} \) |
| 47 | \( 1 + 47iT^{2} \) |
| 53 | \( 1 + (-5 - 5i)T + 53iT^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 12T + 61T^{2} \) |
| 67 | \( 1 + 67iT^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 + (5 + 5i)T + 73iT^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 83iT^{2} \) |
| 89 | \( 1 - 16iT - 89T^{2} \) |
| 97 | \( 1 + (5 - 5i)T - 97iT^{2} \) |
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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
−8.663341063182526701527094196123, −8.343787936092502518486742741931, −7.20841322165309643346451399232, −6.43642984351062783792292115482, −6.09465287100598661084068739588, −4.90362493441245472213598200189, −4.11608941416269968587323820585, −3.51594324937396130503208458110, −2.17375340018088977174639121677, −1.41781912695043533181137177002,
0.34372379665582327172777315066, 1.58067441724035612484084937995, 2.81712487936889509534405219078, 3.49116864711240417036772015058, 4.50330332266216728338567063389, 5.33097919294324448673194778447, 6.01329730194925611310776465776, 6.87520859707041080025998222293, 7.50911852802093353448602397607, 8.504915023803546197615460878070
