| L(s) = 1 | + 3-s + 4.06·5-s + 1.32·7-s + 9-s + 5.43·11-s − 4.61·13-s + 4.06·15-s + 0.597·17-s + 6.78·19-s + 1.32·21-s − 1.97·23-s + 11.5·25-s + 27-s + 7.25·29-s − 0.213·31-s + 5.43·33-s + 5.37·35-s − 7.54·37-s − 4.61·39-s − 4.04·41-s − 6.04·43-s + 4.06·45-s − 7.46·47-s − 5.25·49-s + 0.597·51-s + 11.1·53-s + 22.0·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.81·5-s + 0.499·7-s + 0.333·9-s + 1.63·11-s − 1.28·13-s + 1.04·15-s + 0.144·17-s + 1.55·19-s + 0.288·21-s − 0.411·23-s + 2.30·25-s + 0.192·27-s + 1.34·29-s − 0.0383·31-s + 0.945·33-s + 0.908·35-s − 1.24·37-s − 0.739·39-s − 0.631·41-s − 0.921·43-s + 0.606·45-s − 1.08·47-s − 0.750·49-s + 0.0836·51-s + 1.53·53-s + 2.97·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.441432316\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.441432316\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 251 | \( 1 + T \) |
| good | 5 | \( 1 - 4.06T + 5T^{2} \) |
| 7 | \( 1 - 1.32T + 7T^{2} \) |
| 11 | \( 1 - 5.43T + 11T^{2} \) |
| 13 | \( 1 + 4.61T + 13T^{2} \) |
| 17 | \( 1 - 0.597T + 17T^{2} \) |
| 19 | \( 1 - 6.78T + 19T^{2} \) |
| 23 | \( 1 + 1.97T + 23T^{2} \) |
| 29 | \( 1 - 7.25T + 29T^{2} \) |
| 31 | \( 1 + 0.213T + 31T^{2} \) |
| 37 | \( 1 + 7.54T + 37T^{2} \) |
| 41 | \( 1 + 4.04T + 41T^{2} \) |
| 43 | \( 1 + 6.04T + 43T^{2} \) |
| 47 | \( 1 + 7.46T + 47T^{2} \) |
| 53 | \( 1 - 11.1T + 53T^{2} \) |
| 59 | \( 1 - 3.30T + 59T^{2} \) |
| 61 | \( 1 - 2.91T + 61T^{2} \) |
| 67 | \( 1 + 2.17T + 67T^{2} \) |
| 71 | \( 1 - 5.89T + 71T^{2} \) |
| 73 | \( 1 + 13.6T + 73T^{2} \) |
| 79 | \( 1 + 12.3T + 79T^{2} \) |
| 83 | \( 1 + 10.1T + 83T^{2} \) |
| 89 | \( 1 + 0.550T + 89T^{2} \) |
| 97 | \( 1 + 12.0T + 97T^{2} \) |
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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
−8.260996721770157055198579015311, −7.10860781884490678390366748267, −6.82443937504903743992716825126, −5.92454376176695200701215324570, −5.18361261165311827612604463688, −4.63063941957122016597907082049, −3.45924852003869140622199002186, −2.67433509850845138385110753348, −1.76622175068005133839715656221, −1.25069150444273177369495475648,
1.25069150444273177369495475648, 1.76622175068005133839715656221, 2.67433509850845138385110753348, 3.45924852003869140622199002186, 4.63063941957122016597907082049, 5.18361261165311827612604463688, 5.92454376176695200701215324570, 6.82443937504903743992716825126, 7.10860781884490678390366748267, 8.260996721770157055198579015311
