| L(s) = 1 | − 1.64·2-s − 3-s + 0.718·4-s + 1.64·6-s − 1.44·7-s + 2.11·8-s + 9-s + 0.191·11-s − 0.718·12-s − 5.14·13-s + 2.37·14-s − 4.92·16-s + 3.58·17-s − 1.64·18-s + 7.84·19-s + 1.44·21-s − 0.315·22-s − 7.39·23-s − 2.11·24-s + 8.48·26-s − 27-s − 1.03·28-s + 29-s − 7.18·31-s + 3.88·32-s − 0.191·33-s − 5.90·34-s + ⋯ |
| L(s) = 1 | − 1.16·2-s − 0.577·3-s + 0.359·4-s + 0.673·6-s − 0.544·7-s + 0.746·8-s + 0.333·9-s + 0.0576·11-s − 0.207·12-s − 1.42·13-s + 0.635·14-s − 1.23·16-s + 0.869·17-s − 0.388·18-s + 1.80·19-s + 0.314·21-s − 0.0672·22-s − 1.54·23-s − 0.431·24-s + 1.66·26-s − 0.192·27-s − 0.195·28-s + 0.185·29-s − 1.29·31-s + 0.687·32-s − 0.0332·33-s − 1.01·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4827978145\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4827978145\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 29 | \( 1 - T \) |
| good | 2 | \( 1 + 1.64T + 2T^{2} \) |
| 7 | \( 1 + 1.44T + 7T^{2} \) |
| 11 | \( 1 - 0.191T + 11T^{2} \) |
| 13 | \( 1 + 5.14T + 13T^{2} \) |
| 17 | \( 1 - 3.58T + 17T^{2} \) |
| 19 | \( 1 - 7.84T + 19T^{2} \) |
| 23 | \( 1 + 7.39T + 23T^{2} \) |
| 31 | \( 1 + 7.18T + 31T^{2} \) |
| 37 | \( 1 - 8.70T + 37T^{2} \) |
| 41 | \( 1 - 2.33T + 41T^{2} \) |
| 43 | \( 1 - 3.67T + 43T^{2} \) |
| 47 | \( 1 + 9.52T + 47T^{2} \) |
| 53 | \( 1 + 10.9T + 53T^{2} \) |
| 59 | \( 1 - 14.2T + 59T^{2} \) |
| 61 | \( 1 - 8.45T + 61T^{2} \) |
| 67 | \( 1 + 9.96T + 67T^{2} \) |
| 71 | \( 1 + 4.72T + 71T^{2} \) |
| 73 | \( 1 - 9.06T + 73T^{2} \) |
| 79 | \( 1 + 8.78T + 79T^{2} \) |
| 83 | \( 1 + 15.8T + 83T^{2} \) |
| 89 | \( 1 + 11.6T + 89T^{2} \) |
| 97 | \( 1 - 9.85T + 97T^{2} \) |
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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
−9.409590795301012141403926640535, −8.219734998988752636574827333460, −7.54836464253465057287720286816, −7.10045181058122359229682942546, −5.96169586029042691433341179440, −5.20293871628753033906838984968, −4.28043465467541760506844085576, −3.09276178305079768419663398661, −1.79080821953127293385960464746, −0.56006473139472308493909671056,
0.56006473139472308493909671056, 1.79080821953127293385960464746, 3.09276178305079768419663398661, 4.28043465467541760506844085576, 5.20293871628753033906838984968, 5.96169586029042691433341179440, 7.10045181058122359229682942546, 7.54836464253465057287720286816, 8.219734998988752636574827333460, 9.409590795301012141403926640535
