L(s) = 1 | + 3-s + 1.04·7-s + 9-s + 6.28·11-s − 1.00·13-s − 4.69·17-s − 5.97·19-s + 1.04·21-s + 8.05·23-s + 27-s + 6.91·29-s − 9.52·31-s + 6.28·33-s + 7.69·37-s − 1.00·39-s + 1.56·41-s + 9.94·43-s − 4.84·47-s − 5.90·49-s − 4.69·51-s + 2.82·53-s − 5.97·57-s + 4.08·59-s + 3.16·61-s + 1.04·63-s + 3.17·67-s + 8.05·69-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.395·7-s + 0.333·9-s + 1.89·11-s − 0.279·13-s − 1.13·17-s − 1.37·19-s + 0.228·21-s + 1.67·23-s + 0.192·27-s + 1.28·29-s − 1.71·31-s + 1.09·33-s + 1.26·37-s − 0.161·39-s + 0.245·41-s + 1.51·43-s − 0.706·47-s − 0.843·49-s − 0.656·51-s + 0.388·53-s − 0.791·57-s + 0.531·59-s + 0.404·61-s + 0.131·63-s + 0.387·67-s + 0.969·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7500 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.057864064\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.057864064\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 - 1.04T + 7T^{2} \) |
| 11 | \( 1 - 6.28T + 11T^{2} \) |
| 13 | \( 1 + 1.00T + 13T^{2} \) |
| 17 | \( 1 + 4.69T + 17T^{2} \) |
| 19 | \( 1 + 5.97T + 19T^{2} \) |
| 23 | \( 1 - 8.05T + 23T^{2} \) |
| 29 | \( 1 - 6.91T + 29T^{2} \) |
| 31 | \( 1 + 9.52T + 31T^{2} \) |
| 37 | \( 1 - 7.69T + 37T^{2} \) |
| 41 | \( 1 - 1.56T + 41T^{2} \) |
| 43 | \( 1 - 9.94T + 43T^{2} \) |
| 47 | \( 1 + 4.84T + 47T^{2} \) |
| 53 | \( 1 - 2.82T + 53T^{2} \) |
| 59 | \( 1 - 4.08T + 59T^{2} \) |
| 61 | \( 1 - 3.16T + 61T^{2} \) |
| 67 | \( 1 - 3.17T + 67T^{2} \) |
| 71 | \( 1 - 6.50T + 71T^{2} \) |
| 73 | \( 1 + 0.367T + 73T^{2} \) |
| 79 | \( 1 + 3.32T + 79T^{2} \) |
| 83 | \( 1 - 13.8T + 83T^{2} \) |
| 89 | \( 1 + 2.50T + 89T^{2} \) |
| 97 | \( 1 - 0.182T + 97T^{2} \) |
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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
−8.020935698240477378493982967995, −6.95529146790083196940238400797, −6.76772781929826329272207989602, −5.94415650067527949087107177273, −4.80457923655169771997325252861, −4.30079223076247674874992907641, −3.64687266052152763531932865348, −2.60905392912489700875386288981, −1.85300344574785426663291488849, −0.888981798532948323438824504677,
0.888981798532948323438824504677, 1.85300344574785426663291488849, 2.60905392912489700875386288981, 3.64687266052152763531932865348, 4.30079223076247674874992907641, 4.80457923655169771997325252861, 5.94415650067527949087107177273, 6.76772781929826329272207989602, 6.95529146790083196940238400797, 8.020935698240477378493982967995