L(s) = 1 | − 2.69·2-s + 0.269·3-s + 5.23·4-s − 0.725·6-s − 7-s − 8.70·8-s − 2.92·9-s − 3.78·11-s + 1.41·12-s − 1.24·13-s + 2.69·14-s + 12.9·16-s − 5.98·17-s + 7.87·18-s − 3.38·19-s − 0.269·21-s + 10.1·22-s + 23-s − 2.34·24-s + 3.33·26-s − 1.59·27-s − 5.23·28-s − 7.02·29-s + 6.26·31-s − 17.4·32-s − 1.02·33-s + 16.1·34-s + ⋯ |
L(s) = 1 | − 1.90·2-s + 0.155·3-s + 2.61·4-s − 0.296·6-s − 0.377·7-s − 3.07·8-s − 0.975·9-s − 1.14·11-s + 0.407·12-s − 0.344·13-s + 0.718·14-s + 3.23·16-s − 1.45·17-s + 1.85·18-s − 0.775·19-s − 0.0588·21-s + 2.17·22-s + 0.208·23-s − 0.479·24-s + 0.654·26-s − 0.307·27-s − 0.989·28-s − 1.30·29-s + 1.12·31-s − 3.08·32-s − 0.177·33-s + 2.76·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.1709285813\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1709285813\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 - T \) |
good | 2 | \( 1 + 2.69T + 2T^{2} \) |
| 3 | \( 1 - 0.269T + 3T^{2} \) |
| 11 | \( 1 + 3.78T + 11T^{2} \) |
| 13 | \( 1 + 1.24T + 13T^{2} \) |
| 17 | \( 1 + 5.98T + 17T^{2} \) |
| 19 | \( 1 + 3.38T + 19T^{2} \) |
| 29 | \( 1 + 7.02T + 29T^{2} \) |
| 31 | \( 1 - 6.26T + 31T^{2} \) |
| 37 | \( 1 + 4.84T + 37T^{2} \) |
| 41 | \( 1 + 1.78T + 41T^{2} \) |
| 43 | \( 1 + 3.02T + 43T^{2} \) |
| 47 | \( 1 + 3.90T + 47T^{2} \) |
| 53 | \( 1 - 2.47T + 53T^{2} \) |
| 59 | \( 1 + 2.89T + 59T^{2} \) |
| 61 | \( 1 + 10.3T + 61T^{2} \) |
| 67 | \( 1 + 11.4T + 67T^{2} \) |
| 71 | \( 1 - 2.70T + 71T^{2} \) |
| 73 | \( 1 - 14.1T + 73T^{2} \) |
| 79 | \( 1 - 3.31T + 79T^{2} \) |
| 83 | \( 1 + 7.02T + 83T^{2} \) |
| 89 | \( 1 + 1.59T + 89T^{2} \) |
| 97 | \( 1 - 11.6T + 97T^{2} \) |
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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
−8.559239407409102059138369609578, −7.948513044486693948890164492568, −7.24629647226574559624412982872, −6.50212477576003919237924776957, −5.88389878524314177802605112099, −4.82327374701349192620961318618, −3.32154930869442125169186800964, −2.53370932152700804856023121755, −1.92843561491559684228218792398, −0.28680288490728902721378591750,
0.28680288490728902721378591750, 1.92843561491559684228218792398, 2.53370932152700804856023121755, 3.32154930869442125169186800964, 4.82327374701349192620961318618, 5.88389878524314177802605112099, 6.50212477576003919237924776957, 7.24629647226574559624412982872, 7.948513044486693948890164492568, 8.559239407409102059138369609578