L(s) = 1 | − 1.41·3-s + 5-s + 2.82·7-s − 0.999·9-s − 4.82·11-s − 0.585·13-s − 1.41·15-s − 0.828·17-s + 19-s − 4.00·21-s + 4·23-s + 25-s + 5.65·27-s − 4.82·29-s + 6.82·33-s + 2.82·35-s − 1.75·37-s + 0.828·39-s + 4.82·41-s + 2.82·43-s − 0.999·45-s + 8.48·47-s + 1.00·49-s + 1.17·51-s + 1.07·53-s − 4.82·55-s − 1.41·57-s + ⋯ |
L(s) = 1 | − 0.816·3-s + 0.447·5-s + 1.06·7-s − 0.333·9-s − 1.45·11-s − 0.162·13-s − 0.365·15-s − 0.200·17-s + 0.229·19-s − 0.872·21-s + 0.834·23-s + 0.200·25-s + 1.08·27-s − 0.896·29-s + 1.18·33-s + 0.478·35-s − 0.288·37-s + 0.132·39-s + 0.754·41-s + 0.431·43-s − 0.149·45-s + 1.23·47-s + 0.142·49-s + 0.164·51-s + 0.147·53-s − 0.651·55-s − 0.187·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 19 | \( 1 - T \) |
good | 3 | \( 1 + 1.41T + 3T^{2} \) |
| 7 | \( 1 - 2.82T + 7T^{2} \) |
| 11 | \( 1 + 4.82T + 11T^{2} \) |
| 13 | \( 1 + 0.585T + 13T^{2} \) |
| 17 | \( 1 + 0.828T + 17T^{2} \) |
| 23 | \( 1 - 4T + 23T^{2} \) |
| 29 | \( 1 + 4.82T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 1.75T + 37T^{2} \) |
| 41 | \( 1 - 4.82T + 41T^{2} \) |
| 43 | \( 1 - 2.82T + 43T^{2} \) |
| 47 | \( 1 - 8.48T + 47T^{2} \) |
| 53 | \( 1 - 1.07T + 53T^{2} \) |
| 59 | \( 1 - 2.82T + 59T^{2} \) |
| 61 | \( 1 + 9.65T + 61T^{2} \) |
| 67 | \( 1 - 6.58T + 67T^{2} \) |
| 71 | \( 1 + 7.31T + 71T^{2} \) |
| 73 | \( 1 + 16.1T + 73T^{2} \) |
| 79 | \( 1 + 14.8T + 79T^{2} \) |
| 83 | \( 1 + 8T + 83T^{2} \) |
| 89 | \( 1 + 8.82T + 89T^{2} \) |
| 97 | \( 1 - 6.24T + 97T^{2} \) |
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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
−7.55571858546046352382118531745, −7.16886054249533153222984039792, −5.97301020542692230519175696913, −5.59117010678445213645421387771, −4.98811605024256293591925753804, −4.37254084842539379744185203764, −3.01645250443286324263416034929, −2.32700556284505036480282290071, −1.24008545660311617049862534068, 0,
1.24008545660311617049862534068, 2.32700556284505036480282290071, 3.01645250443286324263416034929, 4.37254084842539379744185203764, 4.98811605024256293591925753804, 5.59117010678445213645421387771, 5.97301020542692230519175696913, 7.16886054249533153222984039792, 7.55571858546046352382118531745