| L(s) = 1 | − 0.493·3-s − 4.54·7-s − 2.75·9-s − 4.61·11-s + 5.54·13-s − 7.16·17-s + 1.35·19-s + 2.24·21-s + 23-s + 2.84·27-s − 3.66·29-s − 4.46·31-s + 2.27·33-s − 3.32·37-s − 2.73·39-s + 8.95·41-s − 8.68·43-s − 9.59·47-s + 13.6·49-s + 3.53·51-s − 10.4·53-s − 0.671·57-s − 2.08·59-s − 0.686·61-s + 12.5·63-s − 8.84·67-s − 0.493·69-s + ⋯ |
| L(s) = 1 | − 0.284·3-s − 1.71·7-s − 0.918·9-s − 1.39·11-s + 1.53·13-s − 1.73·17-s + 0.311·19-s + 0.489·21-s + 0.208·23-s + 0.546·27-s − 0.680·29-s − 0.802·31-s + 0.396·33-s − 0.546·37-s − 0.438·39-s + 1.39·41-s − 1.32·43-s − 1.39·47-s + 1.95·49-s + 0.495·51-s − 1.43·53-s − 0.0888·57-s − 0.271·59-s − 0.0878·61-s + 1.57·63-s − 1.08·67-s − 0.0594·69-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
| \(L(1)\) |
\(\approx\) |
\(0.4557145577\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4557145577\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 0.493T + 3T^{2} \) |
| 7 | \( 1 + 4.54T + 7T^{2} \) |
| 11 | \( 1 + 4.61T + 11T^{2} \) |
| 13 | \( 1 - 5.54T + 13T^{2} \) |
| 17 | \( 1 + 7.16T + 17T^{2} \) |
| 19 | \( 1 - 1.35T + 19T^{2} \) |
| 29 | \( 1 + 3.66T + 29T^{2} \) |
| 31 | \( 1 + 4.46T + 31T^{2} \) |
| 37 | \( 1 + 3.32T + 37T^{2} \) |
| 41 | \( 1 - 8.95T + 41T^{2} \) |
| 43 | \( 1 + 8.68T + 43T^{2} \) |
| 47 | \( 1 + 9.59T + 47T^{2} \) |
| 53 | \( 1 + 10.4T + 53T^{2} \) |
| 59 | \( 1 + 2.08T + 59T^{2} \) |
| 61 | \( 1 + 0.686T + 61T^{2} \) |
| 67 | \( 1 + 8.84T + 67T^{2} \) |
| 71 | \( 1 - 15.9T + 71T^{2} \) |
| 73 | \( 1 + 4.44T + 73T^{2} \) |
| 79 | \( 1 + 8.65T + 79T^{2} \) |
| 83 | \( 1 - 4.43T + 83T^{2} \) |
| 89 | \( 1 - 13.2T + 89T^{2} \) |
| 97 | \( 1 - 1.21T + 97T^{2} \) |
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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
−8.466785045861556468103967401423, −7.56134477429821059158895517944, −6.62707449779817363469083192044, −6.19461252040011894195131214956, −5.57169596588705872742425258230, −4.67153628097836746911245874779, −3.48387040464057098809261971948, −3.09653844147057812454510616937, −2.06582535166420688515581146481, −0.35315054677036822372067047101,
0.35315054677036822372067047101, 2.06582535166420688515581146481, 3.09653844147057812454510616937, 3.48387040464057098809261971948, 4.67153628097836746911245874779, 5.57169596588705872742425258230, 6.19461252040011894195131214956, 6.62707449779817363469083192044, 7.56134477429821059158895517944, 8.466785045861556468103967401423
