L(s) = 1 | − 1.28·3-s + 2.42·5-s + 5.14·7-s − 1.34·9-s + 11-s + 4.22·13-s − 3.11·15-s − 4.57·17-s − 5.80·19-s − 6.61·21-s + 1.15·23-s + 0.883·25-s + 5.58·27-s + 1.27·29-s − 8.89·31-s − 1.28·33-s + 12.4·35-s − 1.72·37-s − 5.42·39-s + 6.63·41-s + 9.05·43-s − 3.26·45-s + 1.34·47-s + 19.4·49-s + 5.87·51-s + 1.59·53-s + 2.42·55-s + ⋯ |
L(s) = 1 | − 0.742·3-s + 1.08·5-s + 1.94·7-s − 0.449·9-s + 0.301·11-s + 1.17·13-s − 0.805·15-s − 1.10·17-s − 1.33·19-s − 1.44·21-s + 0.240·23-s + 0.176·25-s + 1.07·27-s + 0.236·29-s − 1.59·31-s − 0.223·33-s + 2.11·35-s − 0.282·37-s − 0.868·39-s + 1.03·41-s + 1.38·43-s − 0.487·45-s + 0.196·47-s + 2.78·49-s + 0.822·51-s + 0.218·53-s + 0.327·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6028 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6028 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.410236582\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.410236582\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 11 | \( 1 - T \) |
| 137 | \( 1 - T \) |
good | 3 | \( 1 + 1.28T + 3T^{2} \) |
| 5 | \( 1 - 2.42T + 5T^{2} \) |
| 7 | \( 1 - 5.14T + 7T^{2} \) |
| 13 | \( 1 - 4.22T + 13T^{2} \) |
| 17 | \( 1 + 4.57T + 17T^{2} \) |
| 19 | \( 1 + 5.80T + 19T^{2} \) |
| 23 | \( 1 - 1.15T + 23T^{2} \) |
| 29 | \( 1 - 1.27T + 29T^{2} \) |
| 31 | \( 1 + 8.89T + 31T^{2} \) |
| 37 | \( 1 + 1.72T + 37T^{2} \) |
| 41 | \( 1 - 6.63T + 41T^{2} \) |
| 43 | \( 1 - 9.05T + 43T^{2} \) |
| 47 | \( 1 - 1.34T + 47T^{2} \) |
| 53 | \( 1 - 1.59T + 53T^{2} \) |
| 59 | \( 1 - 14.3T + 59T^{2} \) |
| 61 | \( 1 - 8.09T + 61T^{2} \) |
| 67 | \( 1 - 5.24T + 67T^{2} \) |
| 71 | \( 1 + 1.75T + 71T^{2} \) |
| 73 | \( 1 + 4.25T + 73T^{2} \) |
| 79 | \( 1 + 3.94T + 79T^{2} \) |
| 83 | \( 1 - 5.18T + 83T^{2} \) |
| 89 | \( 1 + 11.6T + 89T^{2} \) |
| 97 | \( 1 - 18.8T + 97T^{2} \) |
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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
−8.309546398488780700757448623806, −7.26045152974616458756883242683, −6.48747986926175921747196177467, −5.75768116642167672089449783185, −5.44770174753426907007970545740, −4.53709363979839778598983053672, −3.94515653438244789771714340712, −2.35780190067450101170848886258, −1.88070159341712134291958194723, −0.882903134330169567035299444232,
0.882903134330169567035299444232, 1.88070159341712134291958194723, 2.35780190067450101170848886258, 3.94515653438244789771714340712, 4.53709363979839778598983053672, 5.44770174753426907007970545740, 5.75768116642167672089449783185, 6.48747986926175921747196177467, 7.26045152974616458756883242683, 8.309546398488780700757448623806