| L(s) = 1 | + 1.59·3-s − 1.66·7-s − 0.463·9-s − 5.56·11-s + 6.31·13-s + 4.12·17-s − 19-s − 2.64·21-s + 1.82·23-s − 5.51·27-s − 4.08·29-s − 6.61·31-s − 8.86·33-s + 9.66·37-s + 10.0·39-s − 4.61·41-s + 3.75·43-s + 3.85·47-s − 4.23·49-s + 6.56·51-s + 5.24·53-s − 1.59·57-s + 11.5·59-s + 8.02·61-s + 0.770·63-s − 0.155·67-s + 2.90·69-s + ⋯ |
| L(s) = 1 | + 0.919·3-s − 0.628·7-s − 0.154·9-s − 1.67·11-s + 1.75·13-s + 0.999·17-s − 0.229·19-s − 0.577·21-s + 0.380·23-s − 1.06·27-s − 0.759·29-s − 1.18·31-s − 1.54·33-s + 1.58·37-s + 1.61·39-s − 0.720·41-s + 0.572·43-s + 0.562·47-s − 0.604·49-s + 0.919·51-s + 0.720·53-s − 0.210·57-s + 1.50·59-s + 1.02·61-s + 0.0970·63-s − 0.0189·67-s + 0.349·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.286410256\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.286410256\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 - 1.59T + 3T^{2} \) |
| 7 | \( 1 + 1.66T + 7T^{2} \) |
| 11 | \( 1 + 5.56T + 11T^{2} \) |
| 13 | \( 1 - 6.31T + 13T^{2} \) |
| 17 | \( 1 - 4.12T + 17T^{2} \) |
| 23 | \( 1 - 1.82T + 23T^{2} \) |
| 29 | \( 1 + 4.08T + 29T^{2} \) |
| 31 | \( 1 + 6.61T + 31T^{2} \) |
| 37 | \( 1 - 9.66T + 37T^{2} \) |
| 41 | \( 1 + 4.61T + 41T^{2} \) |
| 43 | \( 1 - 3.75T + 43T^{2} \) |
| 47 | \( 1 - 3.85T + 47T^{2} \) |
| 53 | \( 1 - 5.24T + 53T^{2} \) |
| 59 | \( 1 - 11.5T + 59T^{2} \) |
| 61 | \( 1 - 8.02T + 61T^{2} \) |
| 67 | \( 1 + 0.155T + 67T^{2} \) |
| 71 | \( 1 - 12.1T + 71T^{2} \) |
| 73 | \( 1 - 0.795T + 73T^{2} \) |
| 79 | \( 1 - 7.18T + 79T^{2} \) |
| 83 | \( 1 + 5.88T + 83T^{2} \) |
| 89 | \( 1 - 18.5T + 89T^{2} \) |
| 97 | \( 1 - 2.77T + 97T^{2} \) |
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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
−7.991714606043179980258599518298, −7.42152116085835114249404831086, −6.48094334219505659988754217719, −5.65664904759729249661536443175, −5.32394287967672033061214186929, −3.95346152405845531835412681406, −3.47313269177716019902152465658, −2.78547272096025697743806000467, −2.02015301710794022246088007399, −0.70283306990231567103521116825,
0.70283306990231567103521116825, 2.02015301710794022246088007399, 2.78547272096025697743806000467, 3.47313269177716019902152465658, 3.95346152405845531835412681406, 5.32394287967672033061214186929, 5.65664904759729249661536443175, 6.48094334219505659988754217719, 7.42152116085835114249404831086, 7.991714606043179980258599518298
