L(s) = 1 | − 2·7-s − 4i·11-s + 6i·13-s − 6·17-s + 4·23-s + 5·25-s − 4i·29-s + 10·31-s − 2i·37-s − 2·41-s − 8i·43-s + 12·47-s − 3·49-s − 12i·53-s − 4i·59-s + ⋯ |
L(s) = 1 | − 0.755·7-s − 1.20i·11-s + 1.66i·13-s − 1.45·17-s + 0.834·23-s + 25-s − 0.742i·29-s + 1.79·31-s − 0.328i·37-s − 0.312·41-s − 1.21i·43-s + 1.75·47-s − 0.428·49-s − 1.64i·53-s − 0.520i·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.384875930\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.384875930\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 - 5T^{2} \) |
| 7 | \( 1 + 2T + 7T^{2} \) |
| 11 | \( 1 + 4iT - 11T^{2} \) |
| 13 | \( 1 - 6iT - 13T^{2} \) |
| 17 | \( 1 + 6T + 17T^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 23 | \( 1 - 4T + 23T^{2} \) |
| 29 | \( 1 + 4iT - 29T^{2} \) |
| 31 | \( 1 - 10T + 31T^{2} \) |
| 37 | \( 1 + 2iT - 37T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 + 8iT - 43T^{2} \) |
| 47 | \( 1 - 12T + 47T^{2} \) |
| 53 | \( 1 + 12iT - 53T^{2} \) |
| 59 | \( 1 + 4iT - 59T^{2} \) |
| 61 | \( 1 - 2iT - 61T^{2} \) |
| 67 | \( 1 - 4iT - 67T^{2} \) |
| 71 | \( 1 + 4T + 71T^{2} \) |
| 73 | \( 1 - 10T + 73T^{2} \) |
| 79 | \( 1 + 6T + 79T^{2} \) |
| 83 | \( 1 + 12iT - 83T^{2} \) |
| 89 | \( 1 - 2T + 89T^{2} \) |
| 97 | \( 1 + 6T + 97T^{2} \) |
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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
−8.818442447272362450867433742920, −8.467787197629627861191951810615, −7.07818237263446246175161661843, −6.64029392073483037905432224881, −5.97295710103875823654212043959, −4.81069062198528587242616472805, −4.06396230468058290583962009579, −3.07563218131785714723678892280, −2.13107839483728461906007571724, −0.58923055414970614303377240373,
0.972974583802867190111204081286, 2.53484838795304958687526196288, 3.13160838501214896996928042036, 4.42424876166740720498172238122, 5.00550636684972483533667264911, 6.10083053128858053007161866083, 6.79656106667538285156952172592, 7.46618962569058683889751374971, 8.395300559329333244539484924573, 9.110811305845193024187413671908