L(s) = 1 | + 2.84·3-s − 2.72·5-s − 3.71·7-s + 5.09·9-s − 3.83·11-s + 2.85·13-s − 7.74·15-s + 8.15·17-s + 3.41·19-s − 10.5·21-s + 0.915·23-s + 2.40·25-s + 5.96·27-s − 9.16·29-s + 5.69·31-s − 10.9·33-s + 10.1·35-s − 8.85·37-s + 8.11·39-s − 10.6·41-s − 11.1·43-s − 13.8·45-s − 3.44·47-s + 6.81·49-s + 23.2·51-s − 4.95·53-s + 10.4·55-s + ⋯ |
L(s) = 1 | + 1.64·3-s − 1.21·5-s − 1.40·7-s + 1.69·9-s − 1.15·11-s + 0.790·13-s − 1.99·15-s + 1.97·17-s + 0.783·19-s − 2.30·21-s + 0.190·23-s + 0.480·25-s + 1.14·27-s − 1.70·29-s + 1.02·31-s − 1.90·33-s + 1.70·35-s − 1.45·37-s + 1.29·39-s − 1.66·41-s − 1.70·43-s − 2.06·45-s − 0.502·47-s + 0.973·49-s + 3.24·51-s − 0.680·53-s + 1.40·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 503 | \( 1 - T \) |
good | 3 | \( 1 - 2.84T + 3T^{2} \) |
| 5 | \( 1 + 2.72T + 5T^{2} \) |
| 7 | \( 1 + 3.71T + 7T^{2} \) |
| 11 | \( 1 + 3.83T + 11T^{2} \) |
| 13 | \( 1 - 2.85T + 13T^{2} \) |
| 17 | \( 1 - 8.15T + 17T^{2} \) |
| 19 | \( 1 - 3.41T + 19T^{2} \) |
| 23 | \( 1 - 0.915T + 23T^{2} \) |
| 29 | \( 1 + 9.16T + 29T^{2} \) |
| 31 | \( 1 - 5.69T + 31T^{2} \) |
| 37 | \( 1 + 8.85T + 37T^{2} \) |
| 41 | \( 1 + 10.6T + 41T^{2} \) |
| 43 | \( 1 + 11.1T + 43T^{2} \) |
| 47 | \( 1 + 3.44T + 47T^{2} \) |
| 53 | \( 1 + 4.95T + 53T^{2} \) |
| 59 | \( 1 - 2.81T + 59T^{2} \) |
| 61 | \( 1 + 3.59T + 61T^{2} \) |
| 67 | \( 1 + 9.01T + 67T^{2} \) |
| 71 | \( 1 + 7.15T + 71T^{2} \) |
| 73 | \( 1 + 7.82T + 73T^{2} \) |
| 79 | \( 1 - 13.4T + 79T^{2} \) |
| 83 | \( 1 + 13.5T + 83T^{2} \) |
| 89 | \( 1 - 6.47T + 89T^{2} \) |
| 97 | \( 1 - 16.0T + 97T^{2} \) |
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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
−8.149126616119505455961131044209, −7.50410657750930186850411577059, −7.03443851101408264746127337620, −5.87838102070705400986593584429, −4.94286827515578800462800249655, −3.63262427732308138071765112433, −3.38747366051891895781715916073, −2.99911409126542859038978359923, −1.56773162239517170702300350727, 0,
1.56773162239517170702300350727, 2.99911409126542859038978359923, 3.38747366051891895781715916073, 3.63262427732308138071765112433, 4.94286827515578800462800249655, 5.87838102070705400986593584429, 7.03443851101408264746127337620, 7.50410657750930186850411577059, 8.149126616119505455961131044209