| L(s) = 1 | − 3.11·3-s + 3.70·5-s − 0.546·7-s + 6.70·9-s + 3.66·11-s + 13-s − 11.5·15-s + 5.70·17-s − 2.56·19-s + 1.70·21-s + 6.22·23-s + 8.70·25-s − 11.5·27-s + 2·29-s + 4.75·31-s − 11.4·33-s − 2.02·35-s − 0.298·37-s − 3.11·39-s − 1.40·41-s + 12.6·43-s + 24.8·45-s + 8.96·47-s − 6.70·49-s − 17.7·51-s − 11.4·53-s + 13.5·55-s + ⋯ |
| L(s) = 1 | − 1.79·3-s + 1.65·5-s − 0.206·7-s + 2.23·9-s + 1.10·11-s + 0.277·13-s − 2.97·15-s + 1.38·17-s − 0.589·19-s + 0.371·21-s + 1.29·23-s + 1.74·25-s − 2.21·27-s + 0.371·29-s + 0.853·31-s − 1.98·33-s − 0.341·35-s − 0.0490·37-s − 0.498·39-s − 0.219·41-s + 1.92·43-s + 3.69·45-s + 1.30·47-s − 0.957·49-s − 2.48·51-s − 1.56·53-s + 1.82·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3328 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3328 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.713839116\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.713839116\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 3 | \( 1 + 3.11T + 3T^{2} \) |
| 5 | \( 1 - 3.70T + 5T^{2} \) |
| 7 | \( 1 + 0.546T + 7T^{2} \) |
| 11 | \( 1 - 3.66T + 11T^{2} \) |
| 17 | \( 1 - 5.70T + 17T^{2} \) |
| 19 | \( 1 + 2.56T + 19T^{2} \) |
| 23 | \( 1 - 6.22T + 23T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 - 4.75T + 31T^{2} \) |
| 37 | \( 1 + 0.298T + 37T^{2} \) |
| 41 | \( 1 + 1.40T + 41T^{2} \) |
| 43 | \( 1 - 12.6T + 43T^{2} \) |
| 47 | \( 1 - 8.96T + 47T^{2} \) |
| 53 | \( 1 + 11.4T + 53T^{2} \) |
| 59 | \( 1 + 9.89T + 59T^{2} \) |
| 61 | \( 1 + 61T^{2} \) |
| 67 | \( 1 + 13.1T + 67T^{2} \) |
| 71 | \( 1 + 6.77T + 71T^{2} \) |
| 73 | \( 1 + 5.40T + 73T^{2} \) |
| 79 | \( 1 - 13.5T + 79T^{2} \) |
| 83 | \( 1 + 1.47T + 83T^{2} \) |
| 89 | \( 1 + 1.40T + 89T^{2} \) |
| 97 | \( 1 - 10T + 97T^{2} \) |
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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
−9.009703899823299794153036950336, −7.60040327644134506050698702318, −6.69506223234261420088101503238, −6.20826212317364981559551151975, −5.81015386992286620815371435990, −5.04184366675578037890601778628, −4.30372538607335411763684824654, −2.97633421420419566238429508069, −1.55421736171288589296071946348, −0.977499633894826382732130578315,
0.977499633894826382732130578315, 1.55421736171288589296071946348, 2.97633421420419566238429508069, 4.30372538607335411763684824654, 5.04184366675578037890601778628, 5.81015386992286620815371435990, 6.20826212317364981559551151975, 6.69506223234261420088101503238, 7.60040327644134506050698702318, 9.009703899823299794153036950336
