L(s) = 1 | − 2-s − 0.805·3-s + 4-s + 5-s + 0.805·6-s + 2.46·7-s − 8-s − 2.35·9-s − 10-s + 2.96·11-s − 0.805·12-s + 0.253·13-s − 2.46·14-s − 0.805·15-s + 16-s + 2.99·17-s + 2.35·18-s + 7.75·19-s + 20-s − 1.98·21-s − 2.96·22-s + 5.62·23-s + 0.805·24-s + 25-s − 0.253·26-s + 4.31·27-s + 2.46·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.465·3-s + 0.5·4-s + 0.447·5-s + 0.328·6-s + 0.930·7-s − 0.353·8-s − 0.783·9-s − 0.316·10-s + 0.894·11-s − 0.232·12-s + 0.0703·13-s − 0.657·14-s − 0.208·15-s + 0.250·16-s + 0.725·17-s + 0.554·18-s + 1.78·19-s + 0.223·20-s − 0.432·21-s − 0.632·22-s + 1.17·23-s + 0.164·24-s + 0.200·25-s − 0.0497·26-s + 0.829·27-s + 0.465·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.755236515\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.755236515\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 601 | \( 1 + T \) |
good | 3 | \( 1 + 0.805T + 3T^{2} \) |
| 7 | \( 1 - 2.46T + 7T^{2} \) |
| 11 | \( 1 - 2.96T + 11T^{2} \) |
| 13 | \( 1 - 0.253T + 13T^{2} \) |
| 17 | \( 1 - 2.99T + 17T^{2} \) |
| 19 | \( 1 - 7.75T + 19T^{2} \) |
| 23 | \( 1 - 5.62T + 23T^{2} \) |
| 29 | \( 1 - 2.46T + 29T^{2} \) |
| 31 | \( 1 + 0.0301T + 31T^{2} \) |
| 37 | \( 1 - 4.95T + 37T^{2} \) |
| 41 | \( 1 - 7.67T + 41T^{2} \) |
| 43 | \( 1 + 7.35T + 43T^{2} \) |
| 47 | \( 1 + 6.21T + 47T^{2} \) |
| 53 | \( 1 - 5.55T + 53T^{2} \) |
| 59 | \( 1 + 1.78T + 59T^{2} \) |
| 61 | \( 1 + 4.04T + 61T^{2} \) |
| 67 | \( 1 + 5.65T + 67T^{2} \) |
| 71 | \( 1 + 6.71T + 71T^{2} \) |
| 73 | \( 1 - 11.3T + 73T^{2} \) |
| 79 | \( 1 - 10.3T + 79T^{2} \) |
| 83 | \( 1 + 11.8T + 83T^{2} \) |
| 89 | \( 1 - 4.04T + 89T^{2} \) |
| 97 | \( 1 + 3.40T + 97T^{2} \) |
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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
−8.114521182205163211195271534647, −7.46106580228182674082750171510, −6.70579774488956616253111920131, −5.95729283717903638025850028127, −5.32332910365102513390354348122, −4.70485042783871187601542175039, −3.41837468551648026691756885474, −2.71302052987021725121312804633, −1.46688710851689284115466390565, −0.900976605898024180429601807140,
0.900976605898024180429601807140, 1.46688710851689284115466390565, 2.71302052987021725121312804633, 3.41837468551648026691756885474, 4.70485042783871187601542175039, 5.32332910365102513390354348122, 5.95729283717903638025850028127, 6.70579774488956616253111920131, 7.46106580228182674082750171510, 8.114521182205163211195271534647