L(s) = 1 | − 3-s − 0.309·5-s − 1.15·7-s + 9-s + 5.92·11-s − 4.60·13-s + 0.309·15-s − 7.33·17-s − 3.96·19-s + 1.15·21-s + 23-s − 4.90·25-s − 27-s + 29-s + 7.27·31-s − 5.92·33-s + 0.358·35-s + 2.80·37-s + 4.60·39-s − 4.27·41-s + 5.11·43-s − 0.309·45-s + 3.04·47-s − 5.66·49-s + 7.33·51-s − 13.6·53-s − 1.83·55-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.138·5-s − 0.436·7-s + 0.333·9-s + 1.78·11-s − 1.27·13-s + 0.0800·15-s − 1.77·17-s − 0.909·19-s + 0.252·21-s + 0.208·23-s − 0.980·25-s − 0.192·27-s + 0.185·29-s + 1.30·31-s − 1.03·33-s + 0.0605·35-s + 0.460·37-s + 0.737·39-s − 0.667·41-s + 0.780·43-s − 0.0461·45-s + 0.443·47-s − 0.809·49-s + 1.02·51-s − 1.87·53-s − 0.247·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.028433179\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.028433179\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 - T \) |
good | 5 | \( 1 + 0.309T + 5T^{2} \) |
| 7 | \( 1 + 1.15T + 7T^{2} \) |
| 11 | \( 1 - 5.92T + 11T^{2} \) |
| 13 | \( 1 + 4.60T + 13T^{2} \) |
| 17 | \( 1 + 7.33T + 17T^{2} \) |
| 19 | \( 1 + 3.96T + 19T^{2} \) |
| 31 | \( 1 - 7.27T + 31T^{2} \) |
| 37 | \( 1 - 2.80T + 37T^{2} \) |
| 41 | \( 1 + 4.27T + 41T^{2} \) |
| 43 | \( 1 - 5.11T + 43T^{2} \) |
| 47 | \( 1 - 3.04T + 47T^{2} \) |
| 53 | \( 1 + 13.6T + 53T^{2} \) |
| 59 | \( 1 + 0.903T + 59T^{2} \) |
| 61 | \( 1 - 11.0T + 61T^{2} \) |
| 67 | \( 1 - 7.41T + 67T^{2} \) |
| 71 | \( 1 - 3.44T + 71T^{2} \) |
| 73 | \( 1 + 4.89T + 73T^{2} \) |
| 79 | \( 1 - 14.5T + 79T^{2} \) |
| 83 | \( 1 + 15.6T + 83T^{2} \) |
| 89 | \( 1 - 5.84T + 89T^{2} \) |
| 97 | \( 1 - 5.92T + 97T^{2} \) |
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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
−7.76564240340225091653934756929, −6.73534810060401748764440920765, −6.65387344758839680537395205090, −5.97902970242888345462228564672, −4.82413089449410569251242826658, −4.39308536538691514731378100651, −3.71911977820810570872570728251, −2.55979519168282780603201805756, −1.77226875617993999202612478701, −0.50699869374103678000533268577,
0.50699869374103678000533268577, 1.77226875617993999202612478701, 2.55979519168282780603201805756, 3.71911977820810570872570728251, 4.39308536538691514731378100651, 4.82413089449410569251242826658, 5.97902970242888345462228564672, 6.65387344758839680537395205090, 6.73534810060401748764440920765, 7.76564240340225091653934756929