| L(s) = 1 | + 2.71·3-s − 0.521·5-s + 4.38·9-s − 0.0523·11-s − 5.95·13-s − 1.41·15-s + 8.06·17-s + 4.41·19-s + 5.28·23-s − 4.72·25-s + 3.76·27-s + 4.73·29-s − 8.03·31-s − 0.142·33-s − 5.28·37-s − 16.1·39-s + 41-s + 3.65·43-s − 2.28·45-s + 4.87·47-s + 21.9·51-s − 1.13·53-s + 0.0273·55-s + 11.9·57-s + 5.21·59-s + 10.2·61-s + 3.10·65-s + ⋯ |
| L(s) = 1 | + 1.56·3-s − 0.233·5-s + 1.46·9-s − 0.0157·11-s − 1.65·13-s − 0.365·15-s + 1.95·17-s + 1.01·19-s + 1.10·23-s − 0.945·25-s + 0.725·27-s + 0.879·29-s − 1.44·31-s − 0.0247·33-s − 0.868·37-s − 2.59·39-s + 0.156·41-s + 0.557·43-s − 0.340·45-s + 0.710·47-s + 3.06·51-s − 0.155·53-s + 0.00368·55-s + 1.58·57-s + 0.679·59-s + 1.31·61-s + 0.385·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8036 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8036 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.713922721\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.713922721\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 41 | \( 1 - T \) |
| good | 3 | \( 1 - 2.71T + 3T^{2} \) |
| 5 | \( 1 + 0.521T + 5T^{2} \) |
| 11 | \( 1 + 0.0523T + 11T^{2} \) |
| 13 | \( 1 + 5.95T + 13T^{2} \) |
| 17 | \( 1 - 8.06T + 17T^{2} \) |
| 19 | \( 1 - 4.41T + 19T^{2} \) |
| 23 | \( 1 - 5.28T + 23T^{2} \) |
| 29 | \( 1 - 4.73T + 29T^{2} \) |
| 31 | \( 1 + 8.03T + 31T^{2} \) |
| 37 | \( 1 + 5.28T + 37T^{2} \) |
| 43 | \( 1 - 3.65T + 43T^{2} \) |
| 47 | \( 1 - 4.87T + 47T^{2} \) |
| 53 | \( 1 + 1.13T + 53T^{2} \) |
| 59 | \( 1 - 5.21T + 59T^{2} \) |
| 61 | \( 1 - 10.2T + 61T^{2} \) |
| 67 | \( 1 - 9.37T + 67T^{2} \) |
| 71 | \( 1 - 9.25T + 71T^{2} \) |
| 73 | \( 1 - 15.6T + 73T^{2} \) |
| 79 | \( 1 + 7.46T + 79T^{2} \) |
| 83 | \( 1 - 6.66T + 83T^{2} \) |
| 89 | \( 1 - 15.1T + 89T^{2} \) |
| 97 | \( 1 + 8.41T + 97T^{2} \) |
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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
−7.81014737211440921155303921989, −7.39084917346633678077108153035, −6.85668673910400211265866713501, −5.42176004392672589629142542149, −5.15932638587133987229314416805, −3.95154508392668501063075337046, −3.42443359465262184030506067624, −2.74782839714979358743777684625, −2.02988123158199272133510944566, −0.894181869907808049522342404079,
0.894181869907808049522342404079, 2.02988123158199272133510944566, 2.74782839714979358743777684625, 3.42443359465262184030506067624, 3.95154508392668501063075337046, 5.15932638587133987229314416805, 5.42176004392672589629142542149, 6.85668673910400211265866713501, 7.39084917346633678077108153035, 7.81014737211440921155303921989
