L(s) = 1 | + 3-s − 5-s − 7-s + 9-s − 2.82·11-s + 0.828·13-s − 15-s + 7.65·17-s − 2.82·19-s − 21-s − 4·23-s + 25-s + 27-s − 7.65·29-s − 2.82·31-s − 2.82·33-s + 35-s − 11.6·37-s + 0.828·39-s − 7.65·41-s + 1.65·43-s − 45-s + 11.3·47-s + 49-s + 7.65·51-s + 10.4·53-s + 2.82·55-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.447·5-s − 0.377·7-s + 0.333·9-s − 0.852·11-s + 0.229·13-s − 0.258·15-s + 1.85·17-s − 0.648·19-s − 0.218·21-s − 0.834·23-s + 0.200·25-s + 0.192·27-s − 1.42·29-s − 0.508·31-s − 0.492·33-s + 0.169·35-s − 1.91·37-s + 0.132·39-s − 1.19·41-s + 0.252·43-s − 0.149·45-s + 1.65·47-s + 0.142·49-s + 1.07·51-s + 1.44·53-s + 0.381·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 7 | \( 1 + T \) |
good | 11 | \( 1 + 2.82T + 11T^{2} \) |
| 13 | \( 1 - 0.828T + 13T^{2} \) |
| 17 | \( 1 - 7.65T + 17T^{2} \) |
| 19 | \( 1 + 2.82T + 19T^{2} \) |
| 23 | \( 1 + 4T + 23T^{2} \) |
| 29 | \( 1 + 7.65T + 29T^{2} \) |
| 31 | \( 1 + 2.82T + 31T^{2} \) |
| 37 | \( 1 + 11.6T + 37T^{2} \) |
| 41 | \( 1 + 7.65T + 41T^{2} \) |
| 43 | \( 1 - 1.65T + 43T^{2} \) |
| 47 | \( 1 - 11.3T + 47T^{2} \) |
| 53 | \( 1 - 10.4T + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 9.31T + 61T^{2} \) |
| 67 | \( 1 + 4T + 67T^{2} \) |
| 71 | \( 1 + 10.8T + 71T^{2} \) |
| 73 | \( 1 + 4.82T + 73T^{2} \) |
| 79 | \( 1 + 15.3T + 79T^{2} \) |
| 83 | \( 1 + 17.6T + 83T^{2} \) |
| 89 | \( 1 - 11.6T + 89T^{2} \) |
| 97 | \( 1 + 7.17T + 97T^{2} \) |
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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
−8.341815973234334686735013134223, −7.43087352392658665501795336145, −7.11602385877386513105320175328, −5.78023573032949690179977793912, −5.37674728200089548264560618525, −4.07449901420075033999777888130, −3.54408738565499862211590341845, −2.66424027348982639427096626958, −1.56116966496114657239029463648, 0,
1.56116966496114657239029463648, 2.66424027348982639427096626958, 3.54408738565499862211590341845, 4.07449901420075033999777888130, 5.37674728200089548264560618525, 5.78023573032949690179977793912, 7.11602385877386513105320175328, 7.43087352392658665501795336145, 8.341815973234334686735013134223