| L(s) = 1 | + 3.43·3-s + 2.58·7-s + 8.79·9-s + 1.03·11-s + 1.95·13-s − 1.54·17-s − 2.96·19-s + 8.86·21-s + 23-s + 19.8·27-s + 6.93·29-s − 8.05·31-s + 3.54·33-s + 5.70·37-s + 6.71·39-s − 2.85·41-s − 9.90·43-s + 0.625·47-s − 0.330·49-s − 5.31·51-s − 13.7·53-s − 10.1·57-s + 8.08·59-s + 8.80·61-s + 22.7·63-s − 10.8·67-s + 3.43·69-s + ⋯ |
| L(s) = 1 | + 1.98·3-s + 0.976·7-s + 2.93·9-s + 0.311·11-s + 0.542·13-s − 0.375·17-s − 0.680·19-s + 1.93·21-s + 0.208·23-s + 3.82·27-s + 1.28·29-s − 1.44·31-s + 0.617·33-s + 0.937·37-s + 1.07·39-s − 0.445·41-s − 1.50·43-s + 0.0912·47-s − 0.0472·49-s − 0.744·51-s − 1.88·53-s − 1.34·57-s + 1.05·59-s + 1.12·61-s + 2.86·63-s − 1.32·67-s + 0.413·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.127989940\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.127989940\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 3 | \( 1 - 3.43T + 3T^{2} \) |
| 7 | \( 1 - 2.58T + 7T^{2} \) |
| 11 | \( 1 - 1.03T + 11T^{2} \) |
| 13 | \( 1 - 1.95T + 13T^{2} \) |
| 17 | \( 1 + 1.54T + 17T^{2} \) |
| 19 | \( 1 + 2.96T + 19T^{2} \) |
| 29 | \( 1 - 6.93T + 29T^{2} \) |
| 31 | \( 1 + 8.05T + 31T^{2} \) |
| 37 | \( 1 - 5.70T + 37T^{2} \) |
| 41 | \( 1 + 2.85T + 41T^{2} \) |
| 43 | \( 1 + 9.90T + 43T^{2} \) |
| 47 | \( 1 - 0.625T + 47T^{2} \) |
| 53 | \( 1 + 13.7T + 53T^{2} \) |
| 59 | \( 1 - 8.08T + 59T^{2} \) |
| 61 | \( 1 - 8.80T + 61T^{2} \) |
| 67 | \( 1 + 10.8T + 67T^{2} \) |
| 71 | \( 1 - 2.69T + 71T^{2} \) |
| 73 | \( 1 - 7.84T + 73T^{2} \) |
| 79 | \( 1 - 3.96T + 79T^{2} \) |
| 83 | \( 1 + 9.53T + 83T^{2} \) |
| 89 | \( 1 + 7.25T + 89T^{2} \) |
| 97 | \( 1 - 13.6T + 97T^{2} \) |
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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
−8.252104971667318268265574503844, −7.981815831085874492389639059038, −7.02974953497211071225819125523, −6.46623222388146771909017913081, −5.06721850122944833731566142195, −4.36188417165143386467438697577, −3.70085949095810958095983707761, −2.87276692568220783907152133732, −1.98364144307351388750349097988, −1.34251888739725424912692483510,
1.34251888739725424912692483510, 1.98364144307351388750349097988, 2.87276692568220783907152133732, 3.70085949095810958095983707761, 4.36188417165143386467438697577, 5.06721850122944833731566142195, 6.46623222388146771909017913081, 7.02974953497211071225819125523, 7.981815831085874492389639059038, 8.252104971667318268265574503844
