L(s) = 1 | + 3-s − 2.39·5-s + 1.05·7-s + 9-s + 11-s + 13-s − 2.39·15-s − 5.96·17-s − 5.05·19-s + 1.05·21-s + 9.15·23-s + 0.740·25-s + 27-s − 2.65·29-s − 9.70·31-s + 33-s − 2.51·35-s − 4.51·37-s + 39-s + 3.57·41-s + 7.96·43-s − 2.39·45-s − 1.20·47-s − 5.89·49-s − 5.96·51-s − 0.689·53-s − 2.39·55-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 1.07·5-s + 0.397·7-s + 0.333·9-s + 0.301·11-s + 0.277·13-s − 0.618·15-s − 1.44·17-s − 1.15·19-s + 0.229·21-s + 1.90·23-s + 0.148·25-s + 0.192·27-s − 0.493·29-s − 1.74·31-s + 0.174·33-s − 0.425·35-s − 0.742·37-s + 0.160·39-s + 0.557·41-s + 1.21·43-s − 0.357·45-s − 0.176·47-s − 0.842·49-s − 0.835·51-s − 0.0946·53-s − 0.323·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 - T \) |
good | 5 | \( 1 + 2.39T + 5T^{2} \) |
| 7 | \( 1 - 1.05T + 7T^{2} \) |
| 17 | \( 1 + 5.96T + 17T^{2} \) |
| 19 | \( 1 + 5.05T + 19T^{2} \) |
| 23 | \( 1 - 9.15T + 23T^{2} \) |
| 29 | \( 1 + 2.65T + 29T^{2} \) |
| 31 | \( 1 + 9.70T + 31T^{2} \) |
| 37 | \( 1 + 4.51T + 37T^{2} \) |
| 41 | \( 1 - 3.57T + 41T^{2} \) |
| 43 | \( 1 - 7.96T + 43T^{2} \) |
| 47 | \( 1 + 1.20T + 47T^{2} \) |
| 53 | \( 1 + 0.689T + 53T^{2} \) |
| 59 | \( 1 - 11.4T + 59T^{2} \) |
| 61 | \( 1 + 6.89T + 61T^{2} \) |
| 67 | \( 1 + 14.2T + 67T^{2} \) |
| 71 | \( 1 + 8.10T + 71T^{2} \) |
| 73 | \( 1 - 2.25T + 73T^{2} \) |
| 79 | \( 1 + 11.4T + 79T^{2} \) |
| 83 | \( 1 - 15.4T + 83T^{2} \) |
| 89 | \( 1 + 11.8T + 89T^{2} \) |
| 97 | \( 1 + 14.1T + 97T^{2} \) |
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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
−8.373006004107961159418746894405, −7.38740618740330599299322186473, −7.06726492722977312794255874802, −6.06816143055075228541892629988, −4.92628242478120289019390425791, −4.22223882151821695674478895893, −3.62546153740384014931145410193, −2.58340594849713468473394410657, −1.54662063651924799911171960658, 0,
1.54662063651924799911171960658, 2.58340594849713468473394410657, 3.62546153740384014931145410193, 4.22223882151821695674478895893, 4.92628242478120289019390425791, 6.06816143055075228541892629988, 7.06726492722977312794255874802, 7.38740618740330599299322186473, 8.373006004107961159418746894405