| L(s) = 1 | + 2-s − 3-s − 6-s + 7-s − 8-s + 9-s − 11-s + 13-s + 14-s − 16-s + 17-s + 18-s − 21-s − 22-s + 24-s + 26-s − 27-s + 29-s + 33-s + 34-s − 39-s + 2·41-s − 42-s + 47-s + 48-s − 51-s − 54-s + ⋯ |
| L(s) = 1 | + 2-s − 3-s − 6-s + 7-s − 8-s + 9-s − 11-s + 13-s + 14-s − 16-s + 17-s + 18-s − 21-s − 22-s + 24-s + 26-s − 27-s + 29-s + 33-s + 34-s − 39-s + 2·41-s − 42-s + 47-s + 48-s − 51-s − 54-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.392744903\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.392744903\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 29 | \( 1 - T \) |
| good | 2 | \( 1 - T + T^{2} \) |
| 7 | \( 1 - T + T^{2} \) |
| 11 | \( 1 + T + T^{2} \) |
| 13 | \( 1 - T + T^{2} \) |
| 17 | \( 1 - T + T^{2} \) |
| 19 | \( ( 1 - T )( 1 + T ) \) |
| 23 | \( ( 1 - T )( 1 + T ) \) |
| 31 | \( ( 1 - T )( 1 + T ) \) |
| 37 | \( ( 1 - T )( 1 + T ) \) |
| 41 | \( ( 1 - T )^{2} \) |
| 43 | \( ( 1 - T )( 1 + T ) \) |
| 47 | \( 1 - T + T^{2} \) |
| 53 | \( ( 1 - T )( 1 + T ) \) |
| 59 | \( ( 1 - T )( 1 + T ) \) |
| 61 | \( ( 1 - T )( 1 + T ) \) |
| 67 | \( 1 - T + T^{2} \) |
| 71 | \( ( 1 - T )( 1 + T ) \) |
| 73 | \( ( 1 - T )( 1 + T ) \) |
| 79 | \( ( 1 - T )( 1 + T ) \) |
| 83 | \( ( 1 - T )( 1 + T ) \) |
| 89 | \( 1 + T + T^{2} \) |
| 97 | \( ( 1 - T )( 1 + T ) \) |
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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
−9.321370568322852646004041607217, −8.299521144148710265655763863998, −7.68322310931055212511207218484, −6.59726948489287288956080994924, −5.72870402684193589482400148049, −5.34569751867853074286999359008, −4.54657626068590470003325490890, −3.85187238978307462093102532148, −2.64052181192839054170299244030, −1.11917820683913827346747527289,
1.11917820683913827346747527289, 2.64052181192839054170299244030, 3.85187238978307462093102532148, 4.54657626068590470003325490890, 5.34569751867853074286999359008, 5.72870402684193589482400148049, 6.59726948489287288956080994924, 7.68322310931055212511207218484, 8.299521144148710265655763863998, 9.321370568322852646004041607217
