| L(s) = 1 | − 2.05·2-s + 3.36·3-s + 2.20·4-s − 4.11·5-s − 6.90·6-s − 0.423·8-s + 8.33·9-s + 8.44·10-s + 4.89·11-s + 7.42·12-s − 2.86·13-s − 13.8·15-s − 3.54·16-s − 5.56·17-s − 17.0·18-s + 1.58·19-s − 9.08·20-s − 10.0·22-s + 5.20·23-s − 1.42·24-s + 11.9·25-s + 5.88·26-s + 17.9·27-s − 3.57·29-s + 28.4·30-s − 2.12·31-s + 8.11·32-s + ⋯ |
| L(s) = 1 | − 1.45·2-s + 1.94·3-s + 1.10·4-s − 1.84·5-s − 2.81·6-s − 0.149·8-s + 2.77·9-s + 2.67·10-s + 1.47·11-s + 2.14·12-s − 0.795·13-s − 3.58·15-s − 0.886·16-s − 1.35·17-s − 4.02·18-s + 0.363·19-s − 2.03·20-s − 2.14·22-s + 1.08·23-s − 0.290·24-s + 2.39·25-s + 1.15·26-s + 3.45·27-s − 0.662·29-s + 5.19·30-s − 0.381·31-s + 1.43·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6223 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6223 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.450033323\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.450033323\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 127 | \( 1 - T \) |
| good | 2 | \( 1 + 2.05T + 2T^{2} \) |
| 3 | \( 1 - 3.36T + 3T^{2} \) |
| 5 | \( 1 + 4.11T + 5T^{2} \) |
| 11 | \( 1 - 4.89T + 11T^{2} \) |
| 13 | \( 1 + 2.86T + 13T^{2} \) |
| 17 | \( 1 + 5.56T + 17T^{2} \) |
| 19 | \( 1 - 1.58T + 19T^{2} \) |
| 23 | \( 1 - 5.20T + 23T^{2} \) |
| 29 | \( 1 + 3.57T + 29T^{2} \) |
| 31 | \( 1 + 2.12T + 31T^{2} \) |
| 37 | \( 1 - 4.01T + 37T^{2} \) |
| 41 | \( 1 - 5.33T + 41T^{2} \) |
| 43 | \( 1 + 4.34T + 43T^{2} \) |
| 47 | \( 1 - 3.83T + 47T^{2} \) |
| 53 | \( 1 - 2.13T + 53T^{2} \) |
| 59 | \( 1 - 1.95T + 59T^{2} \) |
| 61 | \( 1 + 12.9T + 61T^{2} \) |
| 67 | \( 1 - 11.8T + 67T^{2} \) |
| 71 | \( 1 + 4.52T + 71T^{2} \) |
| 73 | \( 1 + 8.76T + 73T^{2} \) |
| 79 | \( 1 - 12.4T + 79T^{2} \) |
| 83 | \( 1 + 7.89T + 83T^{2} \) |
| 89 | \( 1 - 0.492T + 89T^{2} \) |
| 97 | \( 1 + 1.80T + 97T^{2} \) |
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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
−8.175831957058190377810902432613, −7.59971696567264427722969640341, −7.08355987609667906645015846409, −6.77622142630963771690505582519, −4.58707985464032030918800526793, −4.25184880426735273918707882577, −3.47928990021192572108288666075, −2.68476890451097529460333615465, −1.72670204379763585323125253114, −0.73418154618201360509712052874,
0.73418154618201360509712052874, 1.72670204379763585323125253114, 2.68476890451097529460333615465, 3.47928990021192572108288666075, 4.25184880426735273918707882577, 4.58707985464032030918800526793, 6.77622142630963771690505582519, 7.08355987609667906645015846409, 7.59971696567264427722969640341, 8.175831957058190377810902432613
