L(s) = 1 | + 0.117·2-s + 0.701·3-s − 1.98·4-s + 0.0823·6-s + 7-s − 0.467·8-s − 2.50·9-s + 5.99·11-s − 1.39·12-s + 5.04·13-s + 0.117·14-s + 3.91·16-s + 0.328·17-s − 0.294·18-s − 1.96·19-s + 0.701·21-s + 0.703·22-s − 23-s − 0.328·24-s + 0.592·26-s − 3.86·27-s − 1.98·28-s + 1.97·29-s − 0.0200·31-s + 1.39·32-s + 4.20·33-s + 0.0385·34-s + ⋯ |
L(s) = 1 | + 0.0829·2-s + 0.404·3-s − 0.993·4-s + 0.0336·6-s + 0.377·7-s − 0.165·8-s − 0.836·9-s + 1.80·11-s − 0.402·12-s + 1.39·13-s + 0.0313·14-s + 0.979·16-s + 0.0795·17-s − 0.0693·18-s − 0.450·19-s + 0.153·21-s + 0.150·22-s − 0.208·23-s − 0.0669·24-s + 0.116·26-s − 0.743·27-s − 0.375·28-s + 0.366·29-s − 0.00359·31-s + 0.246·32-s + 0.732·33-s + 0.00660·34-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
\(L(1)\) |
\(\approx\) |
\(2.044126847\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.044126847\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - 0.117T + 2T^{2} \) |
| 3 | \( 1 - 0.701T + 3T^{2} \) |
| 11 | \( 1 - 5.99T + 11T^{2} \) |
| 13 | \( 1 - 5.04T + 13T^{2} \) |
| 17 | \( 1 - 0.328T + 17T^{2} \) |
| 19 | \( 1 + 1.96T + 19T^{2} \) |
| 29 | \( 1 - 1.97T + 29T^{2} \) |
| 31 | \( 1 + 0.0200T + 31T^{2} \) |
| 37 | \( 1 + 7.44T + 37T^{2} \) |
| 41 | \( 1 - 3.98T + 41T^{2} \) |
| 43 | \( 1 + 6.97T + 43T^{2} \) |
| 47 | \( 1 - 9.47T + 47T^{2} \) |
| 53 | \( 1 - 3.28T + 53T^{2} \) |
| 59 | \( 1 - 7.31T + 59T^{2} \) |
| 61 | \( 1 + 14.8T + 61T^{2} \) |
| 67 | \( 1 + 2.29T + 67T^{2} \) |
| 71 | \( 1 - 3.84T + 71T^{2} \) |
| 73 | \( 1 - 12.7T + 73T^{2} \) |
| 79 | \( 1 - 11.1T + 79T^{2} \) |
| 83 | \( 1 + 3.82T + 83T^{2} \) |
| 89 | \( 1 - 2.02T + 89T^{2} \) |
| 97 | \( 1 - 5.27T + 97T^{2} \) |
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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
−8.527085162205469315388999265661, −8.088718626116588183762528679093, −6.90667752764279677217791321448, −6.12804958754288820806062830574, −5.53303135375186391243959164815, −4.47854009125758025522156173588, −3.81649420478949230271156327937, −3.28107023865208450331457615845, −1.84714825587048724494304475561, −0.842263061726426325634852013846,
0.842263061726426325634852013846, 1.84714825587048724494304475561, 3.28107023865208450331457615845, 3.81649420478949230271156327937, 4.47854009125758025522156173588, 5.53303135375186391243959164815, 6.12804958754288820806062830574, 6.90667752764279677217791321448, 8.088718626116588183762528679093, 8.527085162205469315388999265661