L(s) = 1 | + 0.729·3-s − 3.02·5-s − 2.46·9-s − 2.63·11-s + 0.419·13-s − 2.20·15-s − 4.48·17-s + 4.82·19-s + 0.672·23-s + 4.13·25-s − 3.98·27-s − 7.40·29-s − 3.91·31-s − 1.92·33-s − 2.78·37-s + 0.306·39-s + 41-s + 9.85·43-s + 7.45·45-s − 12.2·47-s − 3.27·51-s − 12.6·53-s + 7.96·55-s + 3.52·57-s + 8.17·59-s − 5.69·61-s − 1.26·65-s + ⋯ |
L(s) = 1 | + 0.421·3-s − 1.35·5-s − 0.822·9-s − 0.794·11-s + 0.116·13-s − 0.569·15-s − 1.08·17-s + 1.10·19-s + 0.140·23-s + 0.826·25-s − 0.767·27-s − 1.37·29-s − 0.703·31-s − 0.334·33-s − 0.458·37-s + 0.0490·39-s + 0.156·41-s + 1.50·43-s + 1.11·45-s − 1.78·47-s − 0.458·51-s − 1.73·53-s + 1.07·55-s + 0.466·57-s + 1.06·59-s − 0.729·61-s − 0.157·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8036 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8036 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7170886853\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7170886853\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 41 | \( 1 - T \) |
good | 3 | \( 1 - 0.729T + 3T^{2} \) |
| 5 | \( 1 + 3.02T + 5T^{2} \) |
| 11 | \( 1 + 2.63T + 11T^{2} \) |
| 13 | \( 1 - 0.419T + 13T^{2} \) |
| 17 | \( 1 + 4.48T + 17T^{2} \) |
| 19 | \( 1 - 4.82T + 19T^{2} \) |
| 23 | \( 1 - 0.672T + 23T^{2} \) |
| 29 | \( 1 + 7.40T + 29T^{2} \) |
| 31 | \( 1 + 3.91T + 31T^{2} \) |
| 37 | \( 1 + 2.78T + 37T^{2} \) |
| 43 | \( 1 - 9.85T + 43T^{2} \) |
| 47 | \( 1 + 12.2T + 47T^{2} \) |
| 53 | \( 1 + 12.6T + 53T^{2} \) |
| 59 | \( 1 - 8.17T + 59T^{2} \) |
| 61 | \( 1 + 5.69T + 61T^{2} \) |
| 67 | \( 1 + 4.13T + 67T^{2} \) |
| 71 | \( 1 - 16.7T + 71T^{2} \) |
| 73 | \( 1 - 11.9T + 73T^{2} \) |
| 79 | \( 1 + 5.02T + 79T^{2} \) |
| 83 | \( 1 - 4.57T + 83T^{2} \) |
| 89 | \( 1 + 12.8T + 89T^{2} \) |
| 97 | \( 1 - 4.57T + 97T^{2} \) |
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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
−7.969666985736344844494663399934, −7.34560496564638799848041231916, −6.61092626853661038009141890014, −5.60527563783350352395784062748, −5.04609553894315134455205788832, −4.12519500012417177425320274195, −3.47137261433355647716801072707, −2.86324929749222902594003691714, −1.89766137736111020700337289517, −0.38822452387893386965980257193,
0.38822452387893386965980257193, 1.89766137736111020700337289517, 2.86324929749222902594003691714, 3.47137261433355647716801072707, 4.12519500012417177425320274195, 5.04609553894315134455205788832, 5.60527563783350352395784062748, 6.61092626853661038009141890014, 7.34560496564638799848041231916, 7.969666985736344844494663399934