L(s) = 1 | − 2.56·3-s − 2.64·5-s − 0.180·7-s + 3.56·9-s + 0.644·11-s + 3.94·13-s + 6.77·15-s + 5.56·17-s − 19-s + 0.463·21-s − 4.46·23-s + 1.99·25-s − 1.43·27-s + 3.94·29-s − 5.48·31-s − 1.65·33-s + 0.478·35-s − 7.48·37-s − 10.1·39-s + 8.41·41-s + 5.76·43-s − 9.41·45-s − 11.2·47-s − 6.96·49-s − 14.2·51-s − 7.58·53-s − 1.70·55-s + ⋯ |
L(s) = 1 | − 1.47·3-s − 1.18·5-s − 0.0683·7-s + 1.18·9-s + 0.194·11-s + 1.09·13-s + 1.74·15-s + 1.35·17-s − 0.229·19-s + 0.101·21-s − 0.930·23-s + 0.398·25-s − 0.276·27-s + 0.733·29-s − 0.985·31-s − 0.287·33-s + 0.0808·35-s − 1.23·37-s − 1.61·39-s + 1.31·41-s + 0.879·43-s − 1.40·45-s − 1.64·47-s − 0.995·49-s − 1.99·51-s − 1.04·53-s − 0.229·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 19 | \( 1 + T \) |
good | 3 | \( 1 + 2.56T + 3T^{2} \) |
| 5 | \( 1 + 2.64T + 5T^{2} \) |
| 7 | \( 1 + 0.180T + 7T^{2} \) |
| 11 | \( 1 - 0.644T + 11T^{2} \) |
| 13 | \( 1 - 3.94T + 13T^{2} \) |
| 17 | \( 1 - 5.56T + 17T^{2} \) |
| 23 | \( 1 + 4.46T + 23T^{2} \) |
| 29 | \( 1 - 3.94T + 29T^{2} \) |
| 31 | \( 1 + 5.48T + 31T^{2} \) |
| 37 | \( 1 + 7.48T + 37T^{2} \) |
| 41 | \( 1 - 8.41T + 41T^{2} \) |
| 43 | \( 1 - 5.76T + 43T^{2} \) |
| 47 | \( 1 + 11.2T + 47T^{2} \) |
| 53 | \( 1 + 7.58T + 53T^{2} \) |
| 59 | \( 1 + 11.8T + 59T^{2} \) |
| 61 | \( 1 - 2.47T + 61T^{2} \) |
| 67 | \( 1 - 8.97T + 67T^{2} \) |
| 71 | \( 1 + 7.63T + 71T^{2} \) |
| 73 | \( 1 - 10.6T + 73T^{2} \) |
| 79 | \( 1 - 10.6T + 79T^{2} \) |
| 83 | \( 1 + 5.65T + 83T^{2} \) |
| 89 | \( 1 + 17.1T + 89T^{2} \) |
| 97 | \( 1 - 3.28T + 97T^{2} \) |
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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
−9.466906830039230609232633670549, −8.273772378712470444575867359796, −7.69529294495516417054308778726, −6.64577625281818495025655608618, −5.98844725780492197174645376794, −5.13931134350098002418834900576, −4.14076036916394777155130239983, −3.37583162159757549011012621570, −1.30931322109709111023474293258, 0,
1.30931322109709111023474293258, 3.37583162159757549011012621570, 4.14076036916394777155130239983, 5.13931134350098002418834900576, 5.98844725780492197174645376794, 6.64577625281818495025655608618, 7.69529294495516417054308778726, 8.273772378712470444575867359796, 9.466906830039230609232633670549