L(s) = 1 | + 0.719·3-s − 5-s + 4.18·7-s − 2.48·9-s − 6.05·13-s − 0.719·15-s + 6.52·17-s + 0.739·19-s + 3.01·21-s − 5.13·23-s + 25-s − 3.94·27-s + 2.08·29-s − 1.12·31-s − 4.18·35-s + 7.11·37-s − 4.35·39-s − 4.02·41-s + 6.29·43-s + 2.48·45-s + 9.24·47-s + 10.5·49-s + 4.69·51-s + 7.77·53-s + 0.532·57-s + 6.16·59-s + 14.6·61-s + ⋯ |
L(s) = 1 | + 0.415·3-s − 0.447·5-s + 1.58·7-s − 0.827·9-s − 1.67·13-s − 0.185·15-s + 1.58·17-s + 0.169·19-s + 0.657·21-s − 1.06·23-s + 0.200·25-s − 0.759·27-s + 0.386·29-s − 0.202·31-s − 0.707·35-s + 1.16·37-s − 0.697·39-s − 0.628·41-s + 0.960·43-s + 0.370·45-s + 1.34·47-s + 1.50·49-s + 0.657·51-s + 1.06·53-s + 0.0704·57-s + 0.802·59-s + 1.88·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.188664232\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.188664232\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 \) |
good | 3 | \( 1 - 0.719T + 3T^{2} \) |
| 7 | \( 1 - 4.18T + 7T^{2} \) |
| 13 | \( 1 + 6.05T + 13T^{2} \) |
| 17 | \( 1 - 6.52T + 17T^{2} \) |
| 19 | \( 1 - 0.739T + 19T^{2} \) |
| 23 | \( 1 + 5.13T + 23T^{2} \) |
| 29 | \( 1 - 2.08T + 29T^{2} \) |
| 31 | \( 1 + 1.12T + 31T^{2} \) |
| 37 | \( 1 - 7.11T + 37T^{2} \) |
| 41 | \( 1 + 4.02T + 41T^{2} \) |
| 43 | \( 1 - 6.29T + 43T^{2} \) |
| 47 | \( 1 - 9.24T + 47T^{2} \) |
| 53 | \( 1 - 7.77T + 53T^{2} \) |
| 59 | \( 1 - 6.16T + 59T^{2} \) |
| 61 | \( 1 - 14.6T + 61T^{2} \) |
| 67 | \( 1 + 5.53T + 67T^{2} \) |
| 71 | \( 1 - 8.87T + 71T^{2} \) |
| 73 | \( 1 + 8.09T + 73T^{2} \) |
| 79 | \( 1 + 0.570T + 79T^{2} \) |
| 83 | \( 1 + 15.2T + 83T^{2} \) |
| 89 | \( 1 + 1.33T + 89T^{2} \) |
| 97 | \( 1 - 9.96T + 97T^{2} \) |
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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
−8.242985070471214916649345344504, −7.55711456119601691044167811655, −7.30116086801218217406061615855, −5.82690164619662873289272726971, −5.35053796568883601512530564803, −4.57590541487655594821437067419, −3.80091879247855101480719487167, −2.72575317476687117032443895497, −2.08825017806877670297369774559, −0.797184245932780702927013193187,
0.797184245932780702927013193187, 2.08825017806877670297369774559, 2.72575317476687117032443895497, 3.80091879247855101480719487167, 4.57590541487655594821437067419, 5.35053796568883601512530564803, 5.82690164619662873289272726971, 7.30116086801218217406061615855, 7.55711456119601691044167811655, 8.242985070471214916649345344504