| L(s) = 1 | + 2-s + 3-s + 4-s + 6-s − 2.37·7-s + 8-s + 9-s + 4.37·11-s + 12-s + 4.37·13-s − 2.37·14-s + 16-s − 4.37·17-s + 18-s + 4.37·19-s − 2.37·21-s + 4.37·22-s + 2.37·23-s + 24-s + 4.37·26-s + 27-s − 2.37·28-s + 6·29-s − 8.74·31-s + 32-s + 4.37·33-s − 4.37·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.408·6-s − 0.896·7-s + 0.353·8-s + 0.333·9-s + 1.31·11-s + 0.288·12-s + 1.21·13-s − 0.634·14-s + 0.250·16-s − 1.06·17-s + 0.235·18-s + 1.00·19-s − 0.517·21-s + 0.932·22-s + 0.494·23-s + 0.204·24-s + 0.857·26-s + 0.192·27-s − 0.448·28-s + 1.11·29-s − 1.57·31-s + 0.176·32-s + 0.761·33-s − 0.749·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.256850830\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.256850830\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 5 | \( 1 \) |
| 37 | \( 1 - T \) |
| good | 7 | \( 1 + 2.37T + 7T^{2} \) |
| 11 | \( 1 - 4.37T + 11T^{2} \) |
| 13 | \( 1 - 4.37T + 13T^{2} \) |
| 17 | \( 1 + 4.37T + 17T^{2} \) |
| 19 | \( 1 - 4.37T + 19T^{2} \) |
| 23 | \( 1 - 2.37T + 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 + 8.74T + 31T^{2} \) |
| 41 | \( 1 + 2.74T + 41T^{2} \) |
| 43 | \( 1 - 4T + 43T^{2} \) |
| 47 | \( 1 + 6.74T + 47T^{2} \) |
| 53 | \( 1 - 0.372T + 53T^{2} \) |
| 59 | \( 1 - 3.25T + 59T^{2} \) |
| 61 | \( 1 - 8.74T + 61T^{2} \) |
| 67 | \( 1 - 8.74T + 67T^{2} \) |
| 71 | \( 1 - 12.7T + 71T^{2} \) |
| 73 | \( 1 + 13.1T + 73T^{2} \) |
| 79 | \( 1 - 4T + 79T^{2} \) |
| 83 | \( 1 + 6.37T + 83T^{2} \) |
| 89 | \( 1 - 7.62T + 89T^{2} \) |
| 97 | \( 1 - 9.48T + 97T^{2} \) |
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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
−8.216228123068362852020176476179, −7.14656901198569158147587521772, −6.67669776249369775278378642322, −6.14977434994634330761303279691, −5.22587016218050467252742844985, −4.25550725738654111998808176906, −3.61076825004540196256521852926, −3.13480480712904013430456127504, −2.02260044003993856344573316923, −1.01904958213642045451321757020,
1.01904958213642045451321757020, 2.02260044003993856344573316923, 3.13480480712904013430456127504, 3.61076825004540196256521852926, 4.25550725738654111998808176906, 5.22587016218050467252742844985, 6.14977434994634330761303279691, 6.67669776249369775278378642322, 7.14656901198569158147587521772, 8.216228123068362852020176476179
