| L(s) = 1 | + (−2.64 + i)2-s − 5.29·3-s + (6.00 − 5.29i)4-s + (14.0 − 5.29i)6-s + 8i·7-s + (−10.5 + 20.0i)8-s + 1.00·9-s − 15.8i·11-s + (−31.7 + 28.0i)12-s − 52.9·13-s + (−8 − 21.1i)14-s + (8.00 − 63.4i)16-s + 14i·17-s + (−2.64 + 1.00i)18-s + 37.0i·19-s + ⋯ |
| L(s) = 1 | + (−0.935 + 0.353i)2-s − 1.01·3-s + (0.750 − 0.661i)4-s + (0.952 − 0.360i)6-s + 0.431i·7-s + (−0.467 + 0.883i)8-s + 0.0370·9-s − 0.435i·11-s + (−0.763 + 0.673i)12-s − 1.12·13-s + (−0.152 − 0.404i)14-s + (0.125 − 0.992i)16-s + 0.199i·17-s + (−0.0346 + 0.0130i)18-s + 0.447i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0230i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0230i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.582299 - 0.00671052i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.582299 - 0.00671052i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (2.64 - i)T \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + 5.29T + 27T^{2} \) |
| 7 | \( 1 - 8iT - 343T^{2} \) |
| 11 | \( 1 + 15.8iT - 1.33e3T^{2} \) |
| 13 | \( 1 + 52.9T + 2.19e3T^{2} \) |
| 17 | \( 1 - 14iT - 4.91e3T^{2} \) |
| 19 | \( 1 - 37.0iT - 6.85e3T^{2} \) |
| 23 | \( 1 + 152iT - 1.21e4T^{2} \) |
| 29 | \( 1 - 158. iT - 2.43e4T^{2} \) |
| 31 | \( 1 - 224T + 2.97e4T^{2} \) |
| 37 | \( 1 - 243.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 70T + 6.89e4T^{2} \) |
| 43 | \( 1 - 439.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 336iT - 1.03e5T^{2} \) |
| 53 | \( 1 + 31.7T + 1.48e5T^{2} \) |
| 59 | \( 1 + 534. iT - 2.05e5T^{2} \) |
| 61 | \( 1 - 95.2iT - 2.26e5T^{2} \) |
| 67 | \( 1 - 174.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 72T + 3.57e5T^{2} \) |
| 73 | \( 1 + 294iT - 3.89e5T^{2} \) |
| 79 | \( 1 - 464T + 4.93e5T^{2} \) |
| 83 | \( 1 - 545.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 266T + 7.04e5T^{2} \) |
| 97 | \( 1 + 994iT - 9.12e5T^{2} \) |
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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
−11.85778396762653848893915762499, −10.87494878091863278085760262947, −10.14810387601231595558451817276, −8.985043243084384283176863868136, −8.021719702271845100832207587415, −6.75140857895597768604712232552, −5.91699982919666314271160818299, −4.92086115989654679130767786062, −2.54315623063400979363637155802, −0.60487308560625746098344417244,
0.77718893750762334175125740961, 2.62055683258117400334377828566, 4.44851949265843549853575812514, 5.88335829941382039452951668664, 7.04834304886679975608279498203, 7.86850817602438211414791684123, 9.334941741811850997805861908116, 10.07302637533515514912591837926, 11.06379514400721045914290222800, 11.77500399686328469370586682519
