| L(s) = 1 | − 2.78·3-s + 2.77·5-s − 7-s + 4.75·9-s − 11-s − 13-s − 7.72·15-s − 2.84·17-s − 6.75·19-s + 2.78·21-s + 0.282·23-s + 2.69·25-s − 4.87·27-s + 8.65·29-s − 2.61·31-s + 2.78·33-s − 2.77·35-s − 2.19·37-s + 2.78·39-s + 9.09·41-s + 6.50·43-s + 13.1·45-s + 0.120·47-s + 49-s + 7.92·51-s + 5.22·53-s − 2.77·55-s + ⋯ |
| L(s) = 1 | − 1.60·3-s + 1.24·5-s − 0.377·7-s + 1.58·9-s − 0.301·11-s − 0.277·13-s − 1.99·15-s − 0.690·17-s − 1.55·19-s + 0.607·21-s + 0.0588·23-s + 0.538·25-s − 0.939·27-s + 1.60·29-s − 0.469·31-s + 0.484·33-s − 0.468·35-s − 0.360·37-s + 0.445·39-s + 1.42·41-s + 0.992·43-s + 1.96·45-s + 0.0176·47-s + 0.142·49-s + 1.10·51-s + 0.718·53-s − 0.373·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9705364144\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9705364144\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| good | 3 | \( 1 + 2.78T + 3T^{2} \) |
| 5 | \( 1 - 2.77T + 5T^{2} \) |
| 17 | \( 1 + 2.84T + 17T^{2} \) |
| 19 | \( 1 + 6.75T + 19T^{2} \) |
| 23 | \( 1 - 0.282T + 23T^{2} \) |
| 29 | \( 1 - 8.65T + 29T^{2} \) |
| 31 | \( 1 + 2.61T + 31T^{2} \) |
| 37 | \( 1 + 2.19T + 37T^{2} \) |
| 41 | \( 1 - 9.09T + 41T^{2} \) |
| 43 | \( 1 - 6.50T + 43T^{2} \) |
| 47 | \( 1 - 0.120T + 47T^{2} \) |
| 53 | \( 1 - 5.22T + 53T^{2} \) |
| 59 | \( 1 - 4.04T + 59T^{2} \) |
| 61 | \( 1 + 9.87T + 61T^{2} \) |
| 67 | \( 1 + 2.78T + 67T^{2} \) |
| 71 | \( 1 + 9.02T + 71T^{2} \) |
| 73 | \( 1 + 9.94T + 73T^{2} \) |
| 79 | \( 1 - 10.1T + 79T^{2} \) |
| 83 | \( 1 + 8.81T + 83T^{2} \) |
| 89 | \( 1 - 17.7T + 89T^{2} \) |
| 97 | \( 1 + 8.81T + 97T^{2} \) |
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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
−8.623928921211787809002377580346, −7.41590186106720844446242658082, −6.65348052649924981724117563696, −6.10600944334416523853602076440, −5.73059457978916894186375750266, −4.79636746124946649397289352309, −4.25946673115780002233438751846, −2.71720456969166454233443933513, −1.86491000695698988415832241966, −0.60199209395898874579751363465,
0.60199209395898874579751363465, 1.86491000695698988415832241966, 2.71720456969166454233443933513, 4.25946673115780002233438751846, 4.79636746124946649397289352309, 5.73059457978916894186375750266, 6.10600944334416523853602076440, 6.65348052649924981724117563696, 7.41590186106720844446242658082, 8.623928921211787809002377580346
