L(s) = 1 | + 1.35·3-s + 3.25·5-s − 0.970·7-s − 1.16·9-s − 13-s + 4.41·15-s − 6.38·17-s − 2.52·19-s − 1.31·21-s − 5.53·23-s + 5.60·25-s − 5.64·27-s − 4.41·29-s + 5.83·31-s − 3.16·35-s + 4.80·37-s − 1.35·39-s − 7.96·41-s + 3.69·43-s − 3.77·45-s − 9.82·47-s − 6.05·49-s − 8.66·51-s + 7.96·53-s − 3.42·57-s − 2.40·59-s − 5.61·61-s + ⋯ |
L(s) = 1 | + 0.783·3-s + 1.45·5-s − 0.366·7-s − 0.386·9-s − 0.277·13-s + 1.14·15-s − 1.54·17-s − 0.578·19-s − 0.287·21-s − 1.15·23-s + 1.12·25-s − 1.08·27-s − 0.819·29-s + 1.04·31-s − 0.534·35-s + 0.790·37-s − 0.217·39-s − 1.24·41-s + 0.564·43-s − 0.563·45-s − 1.43·47-s − 0.865·49-s − 1.21·51-s + 1.09·53-s − 0.453·57-s − 0.312·59-s − 0.719·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6292 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6292 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 \) |
| 13 | \( 1 + T \) |
good | 3 | \( 1 - 1.35T + 3T^{2} \) |
| 5 | \( 1 - 3.25T + 5T^{2} \) |
| 7 | \( 1 + 0.970T + 7T^{2} \) |
| 17 | \( 1 + 6.38T + 17T^{2} \) |
| 19 | \( 1 + 2.52T + 19T^{2} \) |
| 23 | \( 1 + 5.53T + 23T^{2} \) |
| 29 | \( 1 + 4.41T + 29T^{2} \) |
| 31 | \( 1 - 5.83T + 31T^{2} \) |
| 37 | \( 1 - 4.80T + 37T^{2} \) |
| 41 | \( 1 + 7.96T + 41T^{2} \) |
| 43 | \( 1 - 3.69T + 43T^{2} \) |
| 47 | \( 1 + 9.82T + 47T^{2} \) |
| 53 | \( 1 - 7.96T + 53T^{2} \) |
| 59 | \( 1 + 2.40T + 59T^{2} \) |
| 61 | \( 1 + 5.61T + 61T^{2} \) |
| 67 | \( 1 - 7.79T + 67T^{2} \) |
| 71 | \( 1 + 12.4T + 71T^{2} \) |
| 73 | \( 1 - 7.32T + 73T^{2} \) |
| 79 | \( 1 - 6.60T + 79T^{2} \) |
| 83 | \( 1 + 6.60T + 83T^{2} \) |
| 89 | \( 1 + 5.25T + 89T^{2} \) |
| 97 | \( 1 + 9.99T + 97T^{2} \) |
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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
−7.85768420404218348989802590480, −6.79585622632446892042770266180, −6.28827539573482517462608856330, −5.71249673313186096533550521597, −4.80718186049541649568182345838, −3.96841062728950686568216096012, −2.94580403156525885010392510362, −2.28884918886965576652319889610, −1.74884877344926938945122961329, 0,
1.74884877344926938945122961329, 2.28884918886965576652319889610, 2.94580403156525885010392510362, 3.96841062728950686568216096012, 4.80718186049541649568182345838, 5.71249673313186096533550521597, 6.28827539573482517462608856330, 6.79585622632446892042770266180, 7.85768420404218348989802590480