L(s) = 1 | + 0.783·2-s + 3-s − 1.38·4-s + 5-s + 0.783·6-s − 0.845·7-s − 2.65·8-s + 9-s + 0.783·10-s − 3.93·11-s − 1.38·12-s + 1.98·13-s − 0.662·14-s + 15-s + 0.694·16-s + 3.79·17-s + 0.783·18-s − 3.43·19-s − 1.38·20-s − 0.845·21-s − 3.07·22-s + 1.50·23-s − 2.65·24-s + 25-s + 1.55·26-s + 27-s + 1.17·28-s + ⋯ |
L(s) = 1 | + 0.553·2-s + 0.577·3-s − 0.693·4-s + 0.447·5-s + 0.319·6-s − 0.319·7-s − 0.937·8-s + 0.333·9-s + 0.247·10-s − 1.18·11-s − 0.400·12-s + 0.549·13-s − 0.177·14-s + 0.258·15-s + 0.173·16-s + 0.921·17-s + 0.184·18-s − 0.788·19-s − 0.309·20-s − 0.184·21-s − 0.656·22-s + 0.313·23-s − 0.541·24-s + 0.200·25-s + 0.304·26-s + 0.192·27-s + 0.221·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 401 | \( 1 + T \) |
good | 2 | \( 1 - 0.783T + 2T^{2} \) |
| 7 | \( 1 + 0.845T + 7T^{2} \) |
| 11 | \( 1 + 3.93T + 11T^{2} \) |
| 13 | \( 1 - 1.98T + 13T^{2} \) |
| 17 | \( 1 - 3.79T + 17T^{2} \) |
| 19 | \( 1 + 3.43T + 19T^{2} \) |
| 23 | \( 1 - 1.50T + 23T^{2} \) |
| 29 | \( 1 + 2.27T + 29T^{2} \) |
| 31 | \( 1 - 0.964T + 31T^{2} \) |
| 37 | \( 1 - 1.73T + 37T^{2} \) |
| 41 | \( 1 + 8.84T + 41T^{2} \) |
| 43 | \( 1 - 4.05T + 43T^{2} \) |
| 47 | \( 1 - 1.00T + 47T^{2} \) |
| 53 | \( 1 + 12.2T + 53T^{2} \) |
| 59 | \( 1 - 10.3T + 59T^{2} \) |
| 61 | \( 1 - 2.27T + 61T^{2} \) |
| 67 | \( 1 + 14.3T + 67T^{2} \) |
| 71 | \( 1 - 3.18T + 71T^{2} \) |
| 73 | \( 1 + 7.76T + 73T^{2} \) |
| 79 | \( 1 + 11.0T + 79T^{2} \) |
| 83 | \( 1 + 4.24T + 83T^{2} \) |
| 89 | \( 1 - 8.40T + 89T^{2} \) |
| 97 | \( 1 + 5.45T + 97T^{2} \) |
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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
−7.944993634709160597715445878994, −6.96367794029025332129072706937, −6.11045108421944632963419550573, −5.50468117747190862686816592097, −4.83614525920964384695390576783, −4.01682089638259262208842285806, −3.21365466280675246367611901846, −2.65303174382963326608677322129, −1.43511030211021054919235725361, 0,
1.43511030211021054919235725361, 2.65303174382963326608677322129, 3.21365466280675246367611901846, 4.01682089638259262208842285806, 4.83614525920964384695390576783, 5.50468117747190862686816592097, 6.11045108421944632963419550573, 6.96367794029025332129072706937, 7.944993634709160597715445878994