| L(s) = 1 | + 2-s + 3-s + 5-s + 6-s − 8-s + 9-s + 10-s + 15-s − 16-s + 17-s + 18-s − 2·19-s − 23-s − 24-s + 25-s + 27-s + 30-s − 31-s + 34-s − 2·38-s − 40-s + 45-s − 46-s − 47-s − 48-s + 49-s + 50-s + ⋯ |
| L(s) = 1 | + 2-s + 3-s + 5-s + 6-s − 8-s + 9-s + 10-s + 15-s − 16-s + 17-s + 18-s − 2·19-s − 23-s − 24-s + 25-s + 27-s + 30-s − 31-s + 34-s − 2·38-s − 40-s + 45-s − 46-s − 47-s − 48-s + 49-s + 50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(2.592667238\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.592667238\) |
| \(L(1)\) |
|
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 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - T + T^{2} \) |
| 7 | \( ( 1 - T )( 1 + T ) \) |
| 13 | \( ( 1 - T )( 1 + T ) \) |
| 17 | \( 1 - T + T^{2} \) |
| 19 | \( ( 1 + T )^{2} \) |
| 23 | \( 1 + T + T^{2} \) |
| 29 | \( ( 1 - T )( 1 + T ) \) |
| 31 | \( 1 + T + T^{2} \) |
| 37 | \( ( 1 - T )( 1 + T ) \) |
| 41 | \( ( 1 - T )( 1 + T ) \) |
| 43 | \( ( 1 - T )( 1 + T ) \) |
| 47 | \( 1 + T + T^{2} \) |
| 53 | \( 1 + T + T^{2} \) |
| 59 | \( ( 1 - T )( 1 + T ) \) |
| 61 | \( 1 - T + T^{2} \) |
| 67 | \( ( 1 - T )( 1 + T ) \) |
| 71 | \( ( 1 - T )( 1 + T ) \) |
| 73 | \( ( 1 - T )( 1 + T ) \) |
| 79 | \( 1 - T + T^{2} \) |
| 83 | \( ( 1 + T )^{2} \) |
| 89 | \( ( 1 - T )( 1 + T ) \) |
| 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
−9.436501050273085618038975308011, −8.697300444665064658213748402884, −8.051924498767069863714022272454, −6.85909719840929975084904019562, −6.13084077997787846505190216566, −5.34075176233648826951998931411, −4.38449780920399050394267377360, −3.66473119385230487976442090257, −2.67503203988989820020920962402, −1.81312358071115943302850856875,
1.81312358071115943302850856875, 2.67503203988989820020920962402, 3.66473119385230487976442090257, 4.38449780920399050394267377360, 5.34075176233648826951998931411, 6.13084077997787846505190216566, 6.85909719840929975084904019562, 8.051924498767069863714022272454, 8.697300444665064658213748402884, 9.436501050273085618038975308011
