L(s) = 1 | + 1.18·2-s − 3-s − 0.588·4-s − 3.84·5-s − 1.18·6-s − 5.02·7-s − 3.07·8-s + 9-s − 4.56·10-s + 1.17·11-s + 0.588·12-s + 4.56·13-s − 5.96·14-s + 3.84·15-s − 2.47·16-s − 17-s + 1.18·18-s − 5.09·19-s + 2.26·20-s + 5.02·21-s + 1.39·22-s + 0.118·23-s + 3.07·24-s + 9.77·25-s + 5.41·26-s − 27-s + 2.95·28-s + ⋯ |
L(s) = 1 | + 0.840·2-s − 0.577·3-s − 0.294·4-s − 1.71·5-s − 0.485·6-s − 1.89·7-s − 1.08·8-s + 0.333·9-s − 1.44·10-s + 0.354·11-s + 0.169·12-s + 1.26·13-s − 1.59·14-s + 0.992·15-s − 0.619·16-s − 0.242·17-s + 0.280·18-s − 1.16·19-s + 0.505·20-s + 1.09·21-s + 0.297·22-s + 0.0247·23-s + 0.627·24-s + 1.95·25-s + 1.06·26-s − 0.192·27-s + 0.558·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| 157 | \( 1 + T \) |
good | 2 | \( 1 - 1.18T + 2T^{2} \) |
| 5 | \( 1 + 3.84T + 5T^{2} \) |
| 7 | \( 1 + 5.02T + 7T^{2} \) |
| 11 | \( 1 - 1.17T + 11T^{2} \) |
| 13 | \( 1 - 4.56T + 13T^{2} \) |
| 19 | \( 1 + 5.09T + 19T^{2} \) |
| 23 | \( 1 - 0.118T + 23T^{2} \) |
| 29 | \( 1 - 0.483T + 29T^{2} \) |
| 31 | \( 1 - 2.83T + 31T^{2} \) |
| 37 | \( 1 + 0.774T + 37T^{2} \) |
| 41 | \( 1 - 1.91T + 41T^{2} \) |
| 43 | \( 1 - 0.0806T + 43T^{2} \) |
| 47 | \( 1 - 0.641T + 47T^{2} \) |
| 53 | \( 1 - 1.13T + 53T^{2} \) |
| 59 | \( 1 - 14.7T + 59T^{2} \) |
| 61 | \( 1 - 4.40T + 61T^{2} \) |
| 67 | \( 1 - 3.88T + 67T^{2} \) |
| 71 | \( 1 + 2.48T + 71T^{2} \) |
| 73 | \( 1 + 9.15T + 73T^{2} \) |
| 79 | \( 1 - 0.193T + 79T^{2} \) |
| 83 | \( 1 + 6.84T + 83T^{2} \) |
| 89 | \( 1 + 4.55T + 89T^{2} \) |
| 97 | \( 1 + 1.32T + 97T^{2} \) |
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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
−7.15605735982644845260288608436, −6.62403358143349242344295463778, −6.16305420237978726937966457218, −5.41166478045337822887147761968, −4.26480060772434719726171469578, −4.04089387969086226762095396039, −3.47207063280746218936632286928, −2.73891208284787416078554767766, −0.77893689109075508473651328834, 0,
0.77893689109075508473651328834, 2.73891208284787416078554767766, 3.47207063280746218936632286928, 4.04089387969086226762095396039, 4.26480060772434719726171469578, 5.41166478045337822887147761968, 6.16305420237978726937966457218, 6.62403358143349242344295463778, 7.15605735982644845260288608436