L(s) = 1 | + 2-s + 3-s + 4-s − 0.719·5-s + 6-s − 5.05·7-s + 8-s + 9-s − 0.719·10-s − 4.16·11-s + 12-s − 13-s − 5.05·14-s − 0.719·15-s + 16-s − 6.58·17-s + 18-s + 4.38·19-s − 0.719·20-s − 5.05·21-s − 4.16·22-s + 4.96·23-s + 24-s − 4.48·25-s − 26-s + 27-s − 5.05·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.321·5-s + 0.408·6-s − 1.91·7-s + 0.353·8-s + 0.333·9-s − 0.227·10-s − 1.25·11-s + 0.288·12-s − 0.277·13-s − 1.35·14-s − 0.185·15-s + 0.250·16-s − 1.59·17-s + 0.235·18-s + 1.00·19-s − 0.160·20-s − 1.10·21-s − 0.887·22-s + 1.03·23-s + 0.204·24-s − 0.896·25-s − 0.196·26-s + 0.192·27-s − 0.956·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.060677857\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.060677857\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 103 | \( 1 - T \) |
good | 5 | \( 1 + 0.719T + 5T^{2} \) |
| 7 | \( 1 + 5.05T + 7T^{2} \) |
| 11 | \( 1 + 4.16T + 11T^{2} \) |
| 17 | \( 1 + 6.58T + 17T^{2} \) |
| 19 | \( 1 - 4.38T + 19T^{2} \) |
| 23 | \( 1 - 4.96T + 23T^{2} \) |
| 29 | \( 1 + 4.34T + 29T^{2} \) |
| 31 | \( 1 - 1.33T + 31T^{2} \) |
| 37 | \( 1 + 0.572T + 37T^{2} \) |
| 41 | \( 1 - 11.8T + 41T^{2} \) |
| 43 | \( 1 - 0.420T + 43T^{2} \) |
| 47 | \( 1 - 4.13T + 47T^{2} \) |
| 53 | \( 1 - 6.15T + 53T^{2} \) |
| 59 | \( 1 - 8.11T + 59T^{2} \) |
| 61 | \( 1 - 6.14T + 61T^{2} \) |
| 67 | \( 1 - 2.81T + 67T^{2} \) |
| 71 | \( 1 + 1.58T + 71T^{2} \) |
| 73 | \( 1 + 1.70T + 73T^{2} \) |
| 79 | \( 1 - 5.14T + 79T^{2} \) |
| 83 | \( 1 - 9.81T + 83T^{2} \) |
| 89 | \( 1 - 2.78T + 89T^{2} \) |
| 97 | \( 1 - 4.58T + 97T^{2} \) |
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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
−7.47196739210646670797547391376, −7.24577582767818576287926815865, −6.45195476279846249010619208052, −5.74535966616431990095319366896, −5.02603112845384183615599840559, −4.05317224543146516984821627498, −3.54669505080449838035594153472, −2.65844456202739987517925059977, −2.38291511283932283024359529078, −0.57494660254573188590646031201,
0.57494660254573188590646031201, 2.38291511283932283024359529078, 2.65844456202739987517925059977, 3.54669505080449838035594153472, 4.05317224543146516984821627498, 5.02603112845384183615599840559, 5.74535966616431990095319366896, 6.45195476279846249010619208052, 7.24577582767818576287926815865, 7.47196739210646670797547391376