L(s) = 1 | + 3-s − 1.87·7-s + 9-s + 4.15·11-s − 13-s − 17-s − 3.47·19-s − 1.87·21-s + 1.47·23-s + 27-s − 1.80·29-s − 8.96·31-s + 4.15·33-s − 1.72·37-s − 39-s − 8.83·41-s + 4.27·43-s + 1.07·47-s − 3.47·49-s − 51-s + 11.5·53-s − 3.47·57-s − 10.4·59-s − 0.195·61-s − 1.87·63-s − 6.15·67-s + 1.47·69-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.709·7-s + 0.333·9-s + 1.25·11-s − 0.277·13-s − 0.242·17-s − 0.797·19-s − 0.409·21-s + 0.307·23-s + 0.192·27-s − 0.335·29-s − 1.60·31-s + 0.723·33-s − 0.282·37-s − 0.160·39-s − 1.38·41-s + 0.652·43-s + 0.156·47-s − 0.496·49-s − 0.140·51-s + 1.59·53-s − 0.460·57-s − 1.35·59-s − 0.0250·61-s − 0.236·63-s − 0.752·67-s + 0.177·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 \) |
| 13 | \( 1 + T \) |
good | 7 | \( 1 + 1.87T + 7T^{2} \) |
| 11 | \( 1 - 4.15T + 11T^{2} \) |
| 17 | \( 1 + T + 17T^{2} \) |
| 19 | \( 1 + 3.47T + 19T^{2} \) |
| 23 | \( 1 - 1.47T + 23T^{2} \) |
| 29 | \( 1 + 1.80T + 29T^{2} \) |
| 31 | \( 1 + 8.96T + 31T^{2} \) |
| 37 | \( 1 + 1.72T + 37T^{2} \) |
| 41 | \( 1 + 8.83T + 41T^{2} \) |
| 43 | \( 1 - 4.27T + 43T^{2} \) |
| 47 | \( 1 - 1.07T + 47T^{2} \) |
| 53 | \( 1 - 11.5T + 53T^{2} \) |
| 59 | \( 1 + 10.4T + 59T^{2} \) |
| 61 | \( 1 + 0.195T + 61T^{2} \) |
| 67 | \( 1 + 6.15T + 67T^{2} \) |
| 71 | \( 1 - 4.03T + 71T^{2} \) |
| 73 | \( 1 + 1.08T + 73T^{2} \) |
| 79 | \( 1 + 8.27T + 79T^{2} \) |
| 83 | \( 1 - 4.43T + 83T^{2} \) |
| 89 | \( 1 + 1.19T + 89T^{2} \) |
| 97 | \( 1 + 4.24T + 97T^{2} \) |
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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
−7.36645298664567748155529582293, −6.89454872223382602174585629331, −6.26822328567150859116702055305, −5.47172987119443798113192483332, −4.48460188821038210185869096401, −3.81650951315804025123480549992, −3.22527335795665915036373892332, −2.23759444606215977270569269316, −1.41888954097044174978271753810, 0,
1.41888954097044174978271753810, 2.23759444606215977270569269316, 3.22527335795665915036373892332, 3.81650951315804025123480549992, 4.48460188821038210185869096401, 5.47172987119443798113192483332, 6.26822328567150859116702055305, 6.89454872223382602174585629331, 7.36645298664567748155529582293